True or False? In Exercises 43 and 44, determine whether

The Determinant of a Matrix In Exercises 112, find the determinant of the matrix. [1]

The Determinant of a Matrix In Exercises 112, find the determinant of the matrix. [3]

The Determinant of a Matrix In Exercises 112, find the determinant of the matrix. [ 2 3 1 4]

The Determinant of a Matrix In Exercises 112, find the determinant of the matrix. [ 3 5 1 2]

The Determinant of a Matrix In Exercises 112, find the determinant of the matrix. [ 5 6 2 3]

The Determinant of a Matrix In Exercises 112, find the determinant of the matrix. [ 2 4 2 3]

The Determinant of a Matrix In Exercises 112, find the determinant of the matrix. [ 7 1 2 6 3]

The Determinant of a Matrix In Exercises 112, find the determinant of the matrix. [ 1 3 4 5 9

The Determinant of a Matrix In Exercises 112, find the determinant of the matrix. [ 0 0 8 4]

The Determinant of a Matrix In Exercises 112, find the determinant of the matrix. [ 2 6 3 9]

The Determinant of a Matrix In Exercises 112, find the determinant of the matrix. [ 3 4 2 1]

The Determinant of a Matrix In Exercises 112, find the determinant of the matrix. [ 2 4 0 4

Finding the Minors and Cofactors of a Matrix In Exercises 1316, find all (a) minors and (b) cofactors of the matrix. [ 1 3 2 4]

Finding the Minors and Cofactors of a Matrix In Exercises 1316, find all (a) minors and (b) cofactors of the matrix. [ 5 1 6 0]

Finding the Minors and Cofactors of a Matrix In Exercises 1316, find all (a) minors and (b) cofactors of the matrix. [ 3 4 2 2 5 3 1 6 1 ]

Finding the Minors and Cofactors of a Matrix In Exercises 1316, find all (a) minors and (b) cofactors of the matrix. [ 3 6 4 4 3 7 2 1 8 ]

Find the determinant of the matrix in Exercise 15 using the method of expansion by cofactors. Use (a) the second row and (b) the second column.

Find the determinant of the matrix in Exercise 16 using the method of expansion by cofactors. Use (a) the third row and (b) the first column.

Finding a Determinant In Exercises 1932, use expansion by cofactors to find the determinant of the matrix [ 1 3 1 4 2 4 2 0 3 ]

Finding a Determinant In Exercises 1932, use expansion by cofactors to find the determinant of the matrix [ 3 4 2 1 1 0 2 4 1 ]

Finding a Determinant In Exercises 1932, use expansion by cofactors to find the determinant of the matrix [ 2 0 0 4 3 0 6 1 5 ]

Finding a Determinant In Exercises 1932, use expansion by cofactors to find the determinant of the matrix [ 3 7 1 0 11 2 0 0 2 ]

Finding a Determinant In Exercises 1932, use expansion by cofactors to find the determinant of the matrix [ 0.4 0.2 0.3 0.4 0.2 0.2 0.3 0.2 0.2]

Finding a Determinant In Exercises 1932, use expansion by cofactors to find the determinant of the matrix [ 0.1 0.3 0.5 0.2 0.2 0.4 0.3 0.2 0.4]

Finding a Determinant In Exercises 1932, use expansion by cofactors to find the determinant of the matrix [ x 3 1 y 2 1 1 0 1 ]

Finding a Determinant In Exercises 1932, use expansion by cofactors to find the determinant of the matrix [ x 2 1 y 2 5 1 1 1 ]

Finding a Determinant In Exercises 1932, use expansion by cofactors to find the determinant of the matrix [ 5 4 0 0 3 6 2 1 0 4 3 2 6 12 4 2 ]

Finding a Determinant In Exercises 1932, use expansion by cofactors to find the determinant of the matrix [ 3 2 4 1 0 6 1 5 7 11 1 2 0 12 2 10]

Finding a Determinant In Exercises 1932, use expansion by cofactors to find the determinant of the matrix [ w 21 10 40 x 15 24 22 y 24 32 32 z 30 18 35]

Finding a Determinant In Exercises 1932, use expansion by cofactors to find the determinant of the matrix [ w 10 30 30 x 15 20 35 y 25 15 25 z 30 10 40

Finding a Determinant In Exercises 1932, use expansion by cofactors to find the determinant of the matrix [ 5 0 0 0 0 2 1 0 0 0 0 4 2 3 0 0 3 6 4 0 2 2 3 1 2 ]

Finding a Determinant In Exercises 1932, use expansion by cofactors to find the determinant of the matrix [ 4 1 6 0 1 3 2 2 0 4 2 7 5 0 2 1 13 6 0 0 2 12 7 0 9 ]

Finding a Determinant In Exercises 33 and 34, use the method demonstrated in Example 5 to find the determinant of the matrix. [ 3 2 1 0 4 3 4 1 1 ]

Finding a Determinant In Exercises 33 and 34, use the method demonstrated in Example 5 to find the determinant of the matrix. [ 3 0 8 8 5 1 7 4 6 ]

Finding a Determinant In Exercises 3538, use a software program or a graphing utility to find the determinant of the matrix. [ 0.1 0.7 0.1 0.6 0.1 0.3 0.3 0.1 0.8]

Finding a Determinant In Exercises 3538, use a software program or a graphing utility to find the determinant of the matrix. [ 4 1 3 6 3 6 2 1 2 1 4 3 5 2 5 2 ]

Finding a Determinant In Exercises 3538, use a software program or a graphing utility to find the determinant of the matrix. [ 1 0 0 1 2 1 3 2 1 2 2 0 4 2 1 2 ]

Finding a Determinant In Exercises 3538, use a software program or a graphing utility to find the determinant of the matrix. [ 8 1 0 1 2 5 0 8 2 6 1 7 6 5 2 2 1 5 8 0 0 6 3 4 6 ]

Finding the Determinant of a Triangular Matrix In Exercises 3942, find the determinant of the triangular matrix. [ 2 4 3 0 6 7 0 0 2 ]

Finding the Determinant of a Triangular Matrix In Exercises 3942, find the determinant of the triangular matrix. [ 4 0 0 0 7 0 0 0 2 ]

Finding the Determinant of a Triangular Matrix In Exercises 3942, find the determinant of the triangular matrix. [ 5 0 0 0 8 0 0 0 4 6 2 0 2 0 2 1 ]

Finding the Determinant of a Triangular Matrix In Exercises 3942, find the determinant of the triangular matrix. [ 4 1 3 8 0 1 2 5 7 0 0 3 0 0 0 0 2 ]

True or False? In Exercises 43 and 44, determine whether each statement is true or false. If a statement is true, give a reason or cite an appropriate statement from the text. If a statement is false, provide an example that shows the statement is not true in all cases or cite an appropriate statement from the text. (a) The determinant of a 2 2 matrix A is a21a12 a11a22. (b) The determinant of a matrix of order 1 is the entry of the matrix. (c) The ij-cofactor of a square matrix A is the matrix obtained by deleting the ith row and jth column of A.

True or False? In Exercises 43 and 44, determine whether each statement is true or false. If a statement is true, give a reason or cite an appropriate statement from the text. If a statement is false, provide an example that shows the statement is not true in all cases or cite an appropriate statement from the text. (a) To find the determinant of a triangular matrix, add the entries on the main diagonal. (b) To find the determinant of a matrix, expand by cofactors in any row or column. (c) When expanding by cofactors, you need not evaluate the cofactors of zero entries.

Solving an Equation In Exercises 4548, solve for x. x + 3 1 2 x + 2 = 0

Solving an Equation In Exercises 4548, solve for x. x 6 2 3 x + 1 = 0

Solving an Equation In Exercises 4548, solve for x. x 1 3 2 x 2 = 0

Solving an Equation In Exercises 4548, solve for x. x + 3 4 1 x 1 = 0

Solving an Equation In Exercises 4952, find the values of for which the determinant is zero. + 2 1 2

Solving an Equation In Exercises 4952, find the values of for which the determinant is zero. 5 1 3 5

Solving an Equation In Exercises 4952, find the values of for which the determinant is zero. 0 0 2 + 1 1 0 2

Solving an Equation In Exercises 4952, find the values of for which the determinant is zero. 0 2 0 2 1 3 2

Show that the system of linear equations a11x1 a21x1 + + a12x2 a22x2 = = b1 b2 has the solution x1 = b1a22 b2a12 a11a22 a21a12 and x2 = b2a11 b1a21 a11a22 a21a12 when a11a22 a21a12 0.

CAPSTONE For an n n matrix A, explain how to find each value. (a) The minor Mij of the entry aij (b) The cofactor Cij of the entry aij (c) The determinant of A

Entries Involving Expressions In Exercises 5562, evaluate the determinant, in which the entries are functions. Determinants of this type occur when changes of variables are made in calculus. 6u 1 1 3v

Entries Involving Expressions In Exercises 5562, evaluate the determinant, in which the entries are functions. Determinants of this type occur when changes of variables are made in calculus. 3x2 1 3y2 1

Entries Involving Expressions In Exercises 5562, evaluate the determinant, in which the entries are functions. Determinants of this type occur when changes of variables are made in calculus. e2x 2e2x e3x 3e3x

Entries Involving Expressions In Exercises 5562, evaluate the determinant, in which the entries are functions. Determinants of this type occur when changes of variables are made in calculus. ex ex xex (1 x)ex

Entries Involving Expressions In Exercises 5562, evaluate the determinant, in which the entries are functions. Determinants of this type occur when changes of variables are made in calculus. x 1 ln x 1+x

Entries Involving Expressions In Exercises 5562, evaluate the determinant, in which the entries are functions. Determinants of this type occur when changes of variables are made in calculus. x 1 x ln x 1 + ln x

Entries Involving Expressions In Exercises 5562, evaluate the determinant, in which the entries are functions. Determinants of this type occur when changes of variables are made in calculus. cos sin 0 r sin r cos 0 0 0 1

Entries Involving Expressions In Exercises 5562, evaluate the determinant, in which the entries are functions. Determinants of this type occur when changes of variables are made in calculus. 1 v v(1 w) vw u u(1 w) uw 0 uv uv

Verifying an Equation In Exercises 6368, evaluate the determinants to verify the equation. w y x z = y w z x

Verifying an Equation In Exercises 6368, evaluate the determinants to verify the equation. w y cx cz = c w y x z

Verifying an Equation In Exercises 6368, evaluate the determinants to verify the equation. w y x z = w y x + cw z + cy

Verifying an Equation In Exercises 6368, evaluate the determinants to verify the equation. w cw x cx = 0

Verifying an Equation In Exercises 6368, evaluate the determinants to verify the equation. 1 1 1 x y z x2 y2 z2 = (y x)(z x)(z y

Verifying an Equation In Exercises 6368, evaluate the determinants to verify the equation. 1 a a3 1 b b3 1 c c3 = (a b)(b c)(c a)(a + b + c)

You are given the equation x 1 0 0 x 1 c b a = ax2 + bx + c. (a) Verify the equation. (b) Use the equation as a model to find a determinant that is equal to ax3 + bx2 + cx + d.

The determinant of a 2 2 matrix involves two products. The determinant of a 3 3 matrix involves six triple products. Show that the determinant of a 4 4 matrix involves 24 quadruple products

In Exercises 1–12, find the determinant of the matrix. [1]

In Exercises 1–12, find the determinant of the matrix. [?3]

In Exercises 1–12, find the determinant of the matrix. \(\left[\begin{array}{ll}2 & 1 \\ 3 & 4\end{array}\right]\) Text Transcription: [2 & 1 \\ 3 & 4]

In Exercises 1–12, find the determinant of the matrix. \(\left[\begin{array}{rr}-3 & 1 \\ 5 & 2\end{array}\right]\) Text Transcription: [-3 & 1 \\ 5 & 2]

In Exercises 1–12, find the determinant of the matrix. \(\left[\begin{array}{rr}5 & 2 \\ -6 & 3\end{array}\right]\) Text Transcription: [5 & 2 \\ -6 & 3]

In Exercises 1–12, find the determinant of the matrix. \(\left[\begin{array}{rr}2 & -2 \\ 4 & 3\end{array}\right]\) Text Transcription: [2 & -2 \\ 4 & 3]

In Exercises 1–12, find the determinant of the matrix. \(\left[\begin{array}{rr}-7 & 6 \\ \frac{1}{2} & 3\end{array}\right]\) Text Transcription: [-7 & 6 \\ frac 1 2 & 3]

In Exercises 1–12, find the determinant of the matrix. \(\left[\begin{array}{rr}\frac{1}{3} & 5 \\ 4 & -9\end{array}\right]\) Text Transcription: [frac 1 3 & 5 \\ 4 & -9]

In Exercises 1–12, find the determinant of the matrix. \(\left[\begin{array}{ll}0 & 8 \\ 0 & 4\end{array}\right]\) Text Transcription: [0 & 8 \\ 0 & 4]

In Exercises 1–12, find the determinant of the matrix. \(\left[\begin{array}{rr}2 & -3 \\ -6 & 9\end{array}\right]\) Text Transcription: [2 & -3 \\ -6 & 9]

In Exercises 1–12, find the determinant of the matrix. \(\left[\begin{array}{rr}\lambda-3 & 2 \\ 4 & \lambda-1\end{array}\right]\) Text Transcription: [lambda-3 & 2 \\ 4 & \lambda-1\]

In Exercises 1–12, find the determinant of the matrix. \(\left[\begin{array}{rrr}\lambda-2 & 0 \\ 4 & \lambda-4\end{array}\right]\) Text Transcription: [lambda-2 & 0 \\ 4 & \lambda-4]

In Exercises 13–16, find all (a) minors and (b) cofactors of the matrix. \(\left[\begin{array}{ll}1 & 2 \\ 3 & 4\end{array}\right]\) Text Transcription: [1 & 2 \\ 3 & 4\]

In Exercises 13–16, find all (a) minors and (b) cofactors of the matrix. \(\left[\begin{array}{rr}-5 & 6 \\ 1 & 0\end{array}\right]\) Text Transcription: [-5 & 6 \\ 1 & 0]

In Exercises 13–16, find all (a) minors and (b) cofactors of the matrix. \(\left[\begin{array}{rrr}-3 & 2 & 1 \\ 4 & 5 & 6 \\ 2 & -3 & 1\end{array}\right]\) Text Transcription: [-3 & 2 & 1 \\ 4 & 5 & 6 \\ 2 & -3 & 1]

Finding the Minors and Cofactors of a Matrix In Exercises 1316, find all (a) minors and (b) cofactors of the matrix. [ 3 6 4 4 3 7 2 1 8 ]

Find the determinant of the matrix in Exercise 15 using the method of expansion by cofactors. Use (a) the second row and (b) the second column.

Find the determinant of the matrix in Exercise 16 using the method of expansion by cofactors. Use (a) the third row and (b) the first column.

In Exercises 19–32, use expansion by cofactors to find the determinant of the matrix. \(\left[\begin{array}{rrr}1 & 4 & -2 \\ 3 & 2 & 0 \\ -1 & 4 & 3\end{array}\right]\) Text Transcription: [1 & 4 & -2 \\ 3 & 2 & 0 \\ -1 & 4 & 3]

In Exercises 19–32, use expansion by cofactors to find the determinant of the matrix. \(\left[\begin{array}{rrr}3 & -1 & 2 \\ 4 & 1 & 4 \\ -2 & 0 & 1\end{array}\right]\) Text Transcription: [3 & -1 & 2 \\ 4 & 1 & 4 \\ -2 & 0 & 1]

In Exercises 19–32, use expansion by cofactors to find the determinant of the matrix. \(\left[\begin{array}{rrr}2 & 4 & 6 \\ 0 & 3 & 1 \\ 0 & 0 & -5\end{array}\right]\) Text Transcription: [2 & 4 & 6 \\ 0 & 3 & 1 \\ 0 & 0 & -5]

In Exercises 19–32, use expansion by cofactors to find the determinant of the matrix. \(\left[\begin{array}{rrr}-3 & 0 & 0 \\ 7 & 11 & 0 \\ 1 & 2 & 2\end{array}\right]\) Text Transcription: [-3 & 0 & 0 \\ 7 & 11 & 0 \\ 1 & 2 & 2]

In Exercises 19–32, use expansion by cofactors to find the determinant of the matrix. \(\left[\begin{array}{rrr}-0.4 & 0.4 & 0.3 \\ 0.2 & 0.2 & 0.2 \\ 0.3 & 0.2 & 0.2\end{array}\right]\) Text Transcription: [-0.4 & 0.4 & 0.3 \\ 0.2 & 0.2 & 0.2 \\ 0.3 & 0.2 & 0.2]

In Exercises 19–32, use expansion by cofactors to find the determinant of the matrix. \(\left[\begin{array}{rrr}0.1 & 0.2 & 0.3 \\ -0.3 & 0.2 & 0.2 \\ 0.5 & 0.4 & 0.4\end{array}\right]\) Text Transcription: [0.1 & 0.2 & 0.3 \\ -0.3 & 0.2 & 0.2 \\ 0.5 & 0.4 & 0.4]

In Exercises 19–32, use expansion by cofactors to find the determinant of the matrix. \(\left[\begin{array}{rrr}x & y & -1 \\ 3 & 2 & 0 \\ 1 & 1 & 1\end{array}\right]\) Text Transcription: [x & y & -1 \\ 3 & 2 & 0 \\ 1 & 1 & 1]

In Exercises 19–32, use expansion by cofactors to find the determinant of the matrix. \(\left[\begin{array}{rrr}x & y & 1 \\ -2 & -2 & 1 \\ 1 & 5 & 1\end{array}\right]\) Text Transcription: [x & y & 1 \\ -2 & -2 & 1 \\ 1 & 5 & 1]

In Exercises 19–32, use expansion by cofactors to find the determinant of the matrix. \(\left[\begin{array}{rrrr}5 & 3 & 0 & 6 \\ 4 & 6 & 4 & 12 \\ 0 & 2 & -3 & 4 \\ 0 & 1 & -2 & 2\end{array}\right]\) Text Transcription: [5 & 3 & 0 & 6 \\ 4 & 6 & 4 & 12 \\ 0 & 2 & -3 & 4 \\ 0 & 1 & -2 & 2]

In Exercises 19–32, use expansion by cofactors to find the determinant of the matrix. \(\left[\begin{array}{rrrr}3 & 0 & 7 & 0 \\ 2 & 6 & 11 & 12 \\ 4 & 1 & -1 & 2 \\ 1 & 5 & 2 & 10\end{array}\right]\) Text Transcription: [3 & 0 & 7 & 0 \\ 2 & 6 & 11 & 12 \\ 4 & 1 & -1 & 2 \\ 1 & 5 & 2 & 10]

In Exercises 19–32, use expansion by cofactors to find the determinant of the matrix. \(\left[\begin{array}{rrrr}w & x & y & z \\ 21 & -15 & 24 & 30 \\ -10 & 24 & -32 & 18 \\ -40 & 22 & 32 & -35\end{array}\right]\) Text Transcription: [w & x & y & z \\ 21 & -15 & 24 & 30 \\ -10 & 24 & -32 & 18 \\ -40 & 22 & 32 & -35]

In Exercises 19–32, use expansion by cofactors to find the determinant of the matrix. \(\left[\begin{array}{rrrr}w & x & y & z \\ 10 & 15 & -25 & 30 \\ -30 & 20 & -15 & -10 \\ 30 & 35 & -25 & -40\end{array}\right]\) Text Transcription: [w & x & y & z \\ 10 & 15 & -25 & 30 \\ -30 & 20 & -15 & -10 \\ 30 & 35 & -25 & -40]

In Exercises 19–32, use expansion by cofactors to find the determinant of the matrix. \(\left[\begin{array}{rrrrr}5 & 2 & 0 & 0 & -2 \\ 0 & 1 & 4 & 3 & 2 \\ 0 & 0 & 2 & 6 & 3 \\ 0 & 0 & 3 & 4 & 1 \\ 0 & 0 & 0 & 0 & 2\end{array}\right]\) Text Transcription: [5 2 0 0 -2 \\ 0 1 4 3 2 \\ 0 0 2 6 3 \\ 0 0 3 4 1 \\ 0 0 0 0 2\]

In Exercises 19–32, use expansion by cofactors to find the determinant of the matrix. \(\left[\begin{array}{rrrrr}-4 & 3 & 2 & -1 & -2 \\ 1 & -2 & 7 & -13 & -12 \\ -6 & 2 & -5 & -6 & -7 \\ 0 & 0 & 0 & 0 & 0 \\ 1 & -4 & -2 & 0 & -9\end{array}\right]\) Text Transcription: [-4 3 2 -1 -2 \\ 1 -2 7 -13 -12 \\ -6 2 -5 -6 -7 \\ 0 0 0 0 0 \\ 1 -4 -2 0 -9]

In Exercises 33 and 34, use the method demonstrated in Example 5 to find the determinant of the matrix. \(\left[\begin{array}{rrr}3 & 0 & 4 \\ -2 & 4 & 1 \\ 1 & -3 & 1\end{array}\right]\) Text Transcription: [3 & 0 & 4 \\ -2 & 4 & 1 \\ 1 & -3 & 1]

In Exercises 33 and 34, use the method demonstrated in Example 5 to find the determinant of the matrix. \(\left[\begin{array}{rrr}3 & 8 & -7 \\ 0 & -5 & 4 \\ 8 & 1 & 6\end{array}\right]\) Text Transcription: [3 & 8 & -7 \\ 0 & -5 & 4 \\ 8 & 1 & 6]

In Exercises 35–38, use a software program or a graphing utility to find the determinant of the matrix. \(\left[\begin{array}{rrr}0.1 & 0.6 & -0.3 \\ 0.7 & -0.1 & 0.1 \\ 0.1 & 0.3 & -0.8\end{array}\right]\) Text Transcription: [0.1 & 0.6 & -0.3 \\ 0.7 & -0.1 & 0.1 \\ 0.1 & 0.3 & -0.8]

In Exercises 35–38, use a software program or a graphing utility to find the determinant of the matrix. \(\left[\begin{array}{rrrr}4 & 3 & 2 & 5 \\ 1 & 6 & -1 & 2 \\ -3 & 2 & 4 & 5 \\ 6 & 1 & 3 & -2\end{array}\right]\) Text Transcription: [4 & 3 & 2 & 5 \\ 1 & 6 & -1 & 2 \\ -3 & 2 & 4 & 5 \\ 6 & 1 & 3 & -2\]

In Exercises 35–38, use a software program or a graphing utility to find the determinant of the matrix. \(\left[\begin{array}{rrrr}1 & 2 & -1 & 4 \\ 0 & 1 & 2 & -2 \\ 0 & 3 & 2 & -1 \\ 1 & 2 & 0 & -2\end{array}\right]\) Text Transcription: [1 & 2 & -1 & 4 \\ 0 & 1 & 2 & -2 \\ 0 & 3 & 2 & -1 \\ 1 & 2 & 0 & -2]

In Exercises 35–38, use a software program or a graphing utility to find the determinant of the matrix. \(\left[\begin{array}{rrrrr}8 & 5 & 1 & -2 & 0 \\ -1 & 0 & 7 & 1 & 6 \\ 0 & 8 & 6 & 5 & -3 \\ 1 & 2 & 5 & -8 & 4 \\ 2 & 6 & -2 & 0 & 6\end{array}\right]\) Text Transcription: [8 5 1 -2 0 \\ -1 0 7 1 6 \\ 0 8 6 5 -3 \\ 1 2 5 -8 4 \\ 2 6 -2 0 6]

In Exercises 39–42, find the determinant of the triangular matrix. \(\left[\begin{array}{rrr}-2 & 0 & 0 \\ 4 & 6 & 0 \\ -3 & 7 & 2\end{array}\right]\) Text Transcription: [-2 & 0 & 0 \\ 4 & 6 & 0 \\ -3 & 7 & 2]

In Exercises 39–42, find the determinant of the triangular matrix. \(\left[\begin{array}{rrr}4 & 0 & 0 \\ 0 & 7 & 0 \\ 0 & 0 & -2\end{array}\right]\) Text Transcription: [4 & 0 & 0 \\ 0 & 7 & 0 \\ 0 & 0 & -2]

In Exercises 39–42, find the determinant of the triangular matrix. \(\left[\begin{array}{rrrr}5 & 8 & -4 & 2 \\ 0 & 0 & 6 & 0 \\ 0 & 0 & 2 & 2 \\ 0 & 0 & 0 & -1\end{array}\right]\) Text Transcription: [5 & 8 & -4 & 2 \\ 0 & 0 & 6 & 0 \\ 0 & 0 & 2 & 2 \\ 0 & 0 & 0 & -1]

In Exercises 39–42, find the determinant of the triangular matrix. \(\left[\begin{array}{rrrr}4 & 0 & 0 & 0 \\ -1 & \frac{1}{2} & 0 & 0 \\ 3 & 5 & 3 & 0 \\ -8 & 7 & 0 & -2\end{array}\right]\) Text Transcription: [4 & 0 & 0 & 0 \\ -1 & \frac{1}{2} & 0 & 0 \\ 3 & 5 & 3 & 0 \\ -8 & 7 & 0 & -2]

True or False? In Exercises 43 and 44, determine whether each statement is true or false. If a statement is true, give a reason or cite an appropriate statement from the text. If a statement is false, provide an example that shows the statement is not true in all cases or cite an appropriate statement from the text. (a) The determinant of a 2 2 matrix A is a21a12 a11a22. (b) The determinant of a matrix of order 1 is the entry of the matrix. (c) The ij-cofactor of a square matrix A is the matrix obtained by deleting the ith row and jth column of A.

True or False? In Exercises 43 and 44, determine whether each statement is true or false. If a statement is true, give a reason or cite an appropriate statement from the text. If a statement is false, provide an example that shows the statement is not true in all cases or cite an appropriate statement from the text. (a) To find the determinant of a triangular matrix, add the entries on the main diagonal. (b) To find the determinant of a matrix, expand by cofactors in any row or column. (c) When expanding by cofactors, you need not evaluate the cofactors of zero entries.

In Exercises 45–48, solve for x. \(\left|\begin{array}{rr}x+3 & 2 \\ 1 & x+2\end{array}\right|=0\) Text Transcription: |x+3 & 2 \\ 1 & x+2|=0

In Exercises 45–48, solve for x. \(\left|\begin{array}{rr}x-6 & 3 \\ -2 & x+1\end{array}\right|=0\) Text Transcription: |x-6 & 3 \\ -2 & x+1|=0

In Exercises 45–48, solve for x. \(\left|\begin{array}{rr}x-1 & 2 \\ 3 & x-2\end{array}\right|=0\) Text Transcription: |x-1 & 2 \\ 3 & x-2|=0

In Exercises 45–48, solve for x. \(\left|\begin{array}{rr}x+3 & 1 \\ -4 & x-1\end{array}\right|=0\) Text Transcription: |x+3 & 1 \\ -4 & x-1|=0

In Exercises 49-52, find the values of \(\lambda\) for which the determinant is zero. \(\left|\begin{array}{rr}\lambda+2 & 2 \\ 1 & \lambda\end{array}\right|\) Text Transcription: lambda |lambda+2 & 2 \\ 1 & \lambda|

In Exercises 49-52, find the values of \(\lambda\) for which the determinant is zero. \(\left|\begin{array}{rr}\lambda-5 & 3 \\ 1 & \lambda-5\end{array}\right|\) Text Transcription: lambda |lambda-5 & 3 \\ 1 & \lambda-5|

In Exercises 49-52, find the values of \(\lambda\) for which the determinant is zero. \(\left|\begin{array}{rrr}\lambda & 2 & 0 \\ 0 & \lambda+1 & 2 \\ 0 & 1 & \lambda\end{array}\right|\) Text Transcription: lambda |lambda & 2 & 0 \\ 0 & \lambda+1 & 2 \\ 0 & 1 & \lambda|

In Exercises 49-52, find the values of \(\lambda\) for which the determinant is zero. \(\left|\begin{array}{llr}\lambda & 0 & 1 \\ 0 & \lambda & 3 \\ 2 & 2 & \lambda-2\end{array}\right|\) Text Transcription: lambda |lambda & 0 & 1 \\ 0 & \lambda & 3 \\ 2 & 2 & \lambda-2|

Show that the system of linear equations \(a_{11} x_{1}+a_{12} x_{2}=b_{1}\) \(a_{21} x_{1}+a_{22} x_{2}=b_{2}\) has the solution \(x_{1}=\frac{b_{1} a_{22}-b_{2} a_{12}}{a_{11} a_{22}-a_{21} a_{12}} \text { and } \quad x_{2}=\frac{b_{2} a_{11}-b_{1} a_{21}}{a_{11} a_{22}-a_{21} a_{12}}\) when \(a_{11} a_{22}-a_{21} a_{12} \neq 0\). Text Transcription: a_11 x_1+a_12 x_2=b_1 a_21 x_1+a_22 x_2=b_2 x_1 = b1_a_22 ? b_2a_12 a_11a_22 ? a_21a1_2 and x_2 = b_2a_11 ? b_1a_21 a_11a_22 ? a_21a_12 a_11a_22 ? a_21a_12 neq 0

For an \(n \times n\) matrix A, explain how to find each value. (a) The minor \(M_{i j}\) of the entry \(a_{i j}\) (b) The cofactor \(C_{i j}\) of the entry \(a_{i j}\) (c) The determinant of A Text Transcription: n times n M_ij a_ij C_ij a_ij

In Exercises 55–62, evaluate the determinant, in which the entries are functions. Determinants of this type occur when changes of variables are made in calculus. \(\left|\begin{array}{rr}6 u & -1 \\ -1 & 3 v\end{array}\right|\) Text Transcription: |6 u & -1 \\ -1 & 3 v|

In Exercises 55–62, evaluate the determinant, in which the entries are functions. Determinants of this type occur when changes of variables are made in calculus. \(\left|\begin{array}{rr}3 x^{2} & -3 y^{2} \\ 1 & 1\end{array}\right|\) Text Transcription: |3 x^{2} & -3 y^{2} \\ 1 & 1|

In Exercises 55–62, evaluate the determinant, in which the entries are functions. Determinants of this type occur when changes of variables are made in calculus. \(\left|\begin{array}{rr}e^{2 x} & e^{3 x} \\ 2 e^{2 x} & 3 e^{3 x}\end{array}\right|\) Text Transcription: |e^2 x & e^3 x \\ 2 e^2 x & 3 e^3 x|

In Exercises 55–62, evaluate the determinant, in which the entries are functions. Determinants of this type occur when changes of variables are made in calculus. \(\left|\begin{array}{rr}e^{-x} & x e^{-x} \\ -e^{-x} & (1-x) e^{-x}\end{array}\right|\) Text Transcription: |e^-x & x e^-x \\ -e^-x & (1-x) e^-x|

In Exercises 55–62, evaluate the determinant, in which the entries are functions. Determinants of this type occur when changes of variables are made in calculus. \(\left|\begin{array}{ll}x & \ln x \\ 1 & 1 / x\end{array}\right|\) Text Transcription: |x & \ln x \\ 1 & 1 / x|

In Exercises 55–62, evaluate the determinant, in which the entries are functions. Determinants of this type occur when changes of variables are made in calculus. \(\left|\begin{array}{rr}x & x \ln x \\ 1 & 1+\ln x\end{array}\right|\) Text Transcription: |x x ln x \\ 1 1+ln x|

In Exercises 55–62, evaluate the determinant, in which the entries are functions. Determinants of this type occur when changes of variables are made in calculus. \(\left|\begin{array}{rrr}\cos \theta & -r \sin \theta & 0 \\ \sin \theta & r \cos \theta & 0 \\ 0 & 0 & 1\end{array}\right|\) Text Transcription: |cos theta -r sin theta 0 \\ sin theta r cos theta 0 \\ 0 0 1|

In Exercises 55–62, evaluate the determinant, in which the entries are functions. Determinants of this type occur when changes of variables are made in calculus. \(\left|\begin{array}{rrr}1-v & -u & 0 \\ v(1-w) & u(1-w) & -u v \\ v w & u w & u v\end{array}\right|\) Text Transcription: |1-v -u 0 \\ v(1-w) u(1-w) -u v \\ v w u w u v|

In Exercises 63–68, evaluate the determinants to verify the equation. \(\left|\begin{array}{ll}w & x \\ y & z\end{array}\right|=-\left|\begin{array}{ll}y & z \\ w & x\end{array}\right|\) Text Transcription: |w x \\ y z|=-|y z \\ w x|

In Exercises 63–68, evaluate the determinants to verify the equation. \(\left|\begin{array}{rr}w & c x \\ y & c z\end{array}\right|=c\left|\begin{array}{ll}w & x \\ y & z\end{array}\right|\) Text Transcription: |w c x \\ y c z|=c|w x \\ y z|

In Exercises 63–68, evaluate the determinants to verify the equation. \(\left|\begin{array}{ll}w & x \\ y & z\end{array}\right|=\left|\begin{array}{cc}w & x+c w \\ y & z+c y\end{array}\right|\) Text Transcription: |w x \\ y z|=|w x+c w \\ y z+c y\|

In Exercises 63–68, evaluate the determinants to verify the equation. \(\left|\begin{array}{cc}w & x \\ c w & c x\end{array}\right|=0\) Text Transcription: |w & x \\ c w & c x|=0

In Exercises 63–68, evaluate the determinants to verify the equation. \(\left|\begin{array}{ccc}1 & x & x^{2} \\ 1 & y & y^{2} \\ 1 & z & z^{2}\end{array}\right|=(y-x)(z-x)(z-y)\) Text Transcription: |1 x x^2 \\ 1 y y^2 \\ 1 z z^2|=(y-x)(z-x)(z-y)

In Exercises 63–68, evaluate the determinants to verify the equation. \(\left|\begin{array}{rrr}1 & 1 & 1 \\ a & b & c \\ a^{3} & b^{3} & c^{3}\end{array}\right|=(a-b)(b-c)(c-a)(a+b+c)\) Text Transcription: |1 1 1 \\ a b c \\ a^3 b^3 c^3|=(a-b)(b-c)(c-a)(a+b+c)

You are given the equation \(\left|\begin{array}{rrr}x & 0 & c \\ -1 & x & b \\ 0 & -1 & a\end{array}\right|=a x^{2}+b x+c\) (a) Verify the equation. (b) Use the equation as a model to find a determinant that is equal to \(a x^{3}+b x^{2}+c x+d\). Text Transcription: |?1 0 0 x ?1 c b a| = ax^2 + bx + c ax^3 + bx^2 + cx + d

The determinant of a 2 2 matrix involves two products. The determinant of a 3 3 matrix involves six triple products. Show that the determinant of a 4 4 matrix involves 24 quadruple products