Elementary Linear Algebra 12th Edition solutions

Elementary Linear Algebra 12
In Exercises 14, verify that \(\operatorname{det}(A)=\operatorname{det}\left(A^{T}\right)\) \(A=\left[\begin{array}{rr}-2 & 3 \\ 1 & 4\end{array}\right]\) Equation Transcription: Text Transcription: det(A) = det(A^T) A = [-2 & 3 \\ 1 & 4]

Elementary Linear Algebra 12
In Exercises 1-4, verify that \(\operatorname{det}(A)=\operatorname{det}\left(A^{T}\right)\) \(A=\left[\begin{array}{rr}-6 & 1 \\ 2 & -2\end{array}\right]\) Equation Transcription: Text Transcription: det(A) = det(A^T) A = [-6 & 1 \\ 2 & -2]

Elementary Linear Algebra 12
In Exercises 1-4, verify that \(\operatorname{det}(A)=\operatorname{det}\left(A^{T}\right)\) \(A=\left[\begin{array}{rrr}2 & -1 & 3 \\ 1 & 2 & 4 \\ 5 & -3 & 6\end{array}\right]\) Equation Transcription: = [] Text Transcription: det(A) = det(A^T) A = [2 &

Elementary Linear Algebra 12
In Exercises 1-4, verify that \(\operatorname{det}(A)=\operatorname{det}\left(A^{T}\right)\) \(A=\left[\begin{array}{rrr}4 & 2 & -1 \\ 0 & 2 & -3 \\ -1 & 1 & 5\end{array}\right]\) Equation Transcription: = [] Text Transcription: det(A) = det(A^T) A = [

Elementary Linear Algebra 12
In Exercises 58, find the determinant of the given elementary matrix by inspection. \(\left[\begin{array}{rrrr}1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & -5 & 0 \\ 0 & 0 & 0 & 1\end{array}\right]\) Equation Transcription: [] Text Transcription: [1 & 0 & 0 & 0 \\ 0

Elementary Linear Algebra 12
In Exercises 58, find the determinant of the given elementary matrix by inspection. \(\left[\begin{array}{rrr}1 & 0 & 0 \\ 0 & 1 & 0 \\ -5 & 0 & 1\end{array}\right]\) Equation Transcription: [] Text Transcription: [ & 0 & 0 \\ 0 & 1 & 0 \\ -5 & 0 & 1]

Elementary Linear Algebra 12
In Exercises 58, find the determinant of the given elementary matrix by inspection. \(\left[\begin{array}{llll}1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1\end{array}\right]\) Equation Transcription: [] Text Transcription:

Elementary Linear Algebra 12
In Exercises 58, find the determinant of the given elementary matrix by inspection. \(\left[\begin{array}{rrrr}1 & 0 & 0 & 0 \\ 0 & -\frac{1}{3} & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1\end{array}\right]\) Equation Transcription: [] Text Transcription: [1

Elementary Linear Algebra 12
In Exercises 914, evaluate the determinant of the matrix by first reducing the matrix to row echelon form and then using some combination of row operations and cofactor expansion. \(\left[\begin{array}{lll}\ \ 3&-6&\ \ 9\\ -2&\ \ 7&-2\\ \ \ 0&\ \ 1&\ \ 5\end{array}\right]\) Equati

Elementary Linear Algebra 12
In Exercises 914, evaluate the determinant of the matrix by first reducing the matrix to row echelon form and then using some combination of row operations and cofactor expansion. \(\left[\begin{array}{rrr}3 & 6 & -9 \\ 0 & 0 & -2 \\ -2 & 1 & 5\end{array}\right]\) Equation Transcr

Elementary Linear Algebra 12
In Exercises 914, evaluate the determinant of the matrix by first reducing the matrix to row echelon form and then using some combination of row operations and cofactor expansion. \(\left[\begin{array}{llll}2 & 1 & 3 & 1 \\ 1 & 0 & 1 & 1 \\ 0 & 2 & 1 & 0 \\ 0 & 1 & 2 & 3\end{array}\right]\)

Elementary Linear Algebra 12
In Exercises 914, evaluate the determinant of the matrix by first reducing the matrix to row echelon form and then using some combination of row operations and cofactor expansion. \(\left[\begin{array}{rrr}1 & -3 & 0 \\ -2 & 4 & 1 \\ 5 & -2 & 2\end{array}\right]\) Equation Transcrip

Elementary Linear Algebra 12
In Exercises 914, evaluate the determinant of the matrix by first reducing the matrix to row echelon form and then using some combination of row operations and cofactor expansion. \(\left[\begin{array}{rrrrr}1 & 3 & 1 & 5 & 3 \\ -2 & -7 & 0 & -4 & 2 \\ 0 & 0 & 1 & 0 & 1 \\ 0 & 0 & 2 & 1 & 1 \\ 0 &

Elementary Linear Algebra 12
In Exercises 914, evaluate the determinant of the matrix by first reducing the matrix to row echelon form and then using some combination of row operations and cofactor expansion. \(\left[\begin{array}{rrrr}1 & -2 & 3 & 1 \\5 & -9 & 6 & 3 \\-1 & 2 & -6 & -2 \\2 & 8 & 6 & `1 \end{array}\right]\)

Elementary Linear Algebra 12
In Exercises 1522, evaluate the determinant, given that \(\left|\begin{array}{lll}a & b & c \\d & e & f \\g & h & i\end{array}\right|=-6\) \(\left|\begin{array}{lll}d & e & f \\ g & h & i \\ a & b & c\end{array}\right|\)

Elementary Linear Algebra 12
In Exercises 1522, evaluate the determinant, given that \(\left|\begin{array}{lll}a & b & c \\d & e & f \\g & h & i\end{array}\right|=-6\) \(\left|\begin{array}{lll}g & h & i \\ d & e & f \\ a & b & c\end{array}\right|\) Equa

Elementary Linear Algebra 12
In Exercises 1522, evaluate the determinant, given that \(\left|\begin{array}{lll}a & b & c \\d & e & f \\g & h & i\end{array}\right|=-6\) \(\left|\begin{array}{rrr}3 a & 3 b & 3 c \\ -d & -e & -f \\ 4 g & 4 h & 4 i\end{array}\right|\)

Elementary Linear Algebra 12
In Exercises 1522, evaluate the determinant, given that \(\left|\begin{array}{lll}a & b & c \\d & e & f \\g & h & i\end{array}\right|=-6\) \(\left|\begin{array}{ccc}a+d & b+e & c+f \\ -d & -e & -f \\ g & h & i\end{array}\right|\) Equat

Elementary Linear Algebra 12
In Exercises 1522, evaluate the determinant, given that \(\left|\begin{array}{lll}a & b & c \\d & e & f \\g & h & i\end{array}\right|=-6\) \(\left|\begin{array}{ccc}a+g & b+h & c+i \\ d & e & f \\ g & h & i\end{array}\right|\) E

Elementary Linear Algebra 12
In Exercises 1522, evaluate the determinant, given that \(\left|\begin{array}{lll}a & b & c \\d & e & f \\g & h & i\end{array}\right|=-6\) \(\left|\begin{array}{ccc} a & b & c \\ 2 d & 2 e & 2 f \\ g+3 a & h+3 b & i+3 c \end{array}\right|\)

Elementary Linear Algebra 12
In Exercises 1522, evaluate the determinant, given that \(\left|\begin{array}{lll}a & b & c \\d & e & f \\g & h & i\end{array}\right|=-6\) \(\left|\begin{array}{ccc}-3 a & -3 b & -3 c \\ d & e & f \\ g-4 d & h-4 e & i-4 f\end{array}\right|\)

Elementary Linear Algebra 12
In Exercises 1522, evaluate the determinant, given that \(\left|\begin{array}{lll}a & b & c \\d & e & f \\g & h & i\end{array}\right|=-6\) \(\left|\begin{array}{ccc}a & b & c \\ d & e & f \\ 2 a & 2 b & 2 c\end{array}\right|\) Equation

Elementary Linear Algebra 12
Use row reduction to show that \(\left|\begin{array}{lll} 1 & 1 & 1 \\ a & b & c \\ a^{2} & b^{2} & c^{2} \end{array}\right|=(b-a)(c-a)(c-b)\) Equation Transcription: || = Text Transcription: |1 & 1 & 1 \\a & b & c \\a^{2} & b^{2} & c^{2| = (b-a)(c-a)(c-

Elementary Linear Algebra 12
Verify the formulas in parts (a) and (b) and then make a conjecture about a general result of which these results are special cases. a. \(\operatorname{det}\left[\begin{array}{lll}0 & 0 & a_{13} \\ 0 & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33}\end{array}\right]=-a_{13} a_{22} a_{31}\) b. \(

Elementary Linear Algebra 12
In Exercises 2528, confirm the identities without evaluating any of the determinants directly. \(\left|\begin{array}{lll} a_{1} & b_{1} & a_{1}+b_{1}+c_{1} \\ a_{2} & b_{2} & a_{2}+b_{2}+c_{2} \\ a_{3} & b_{3} & a_{3}+b_{3}+c_{3} \end{array}\right|=\left|\begin{array}{lll} a_{1} & b_{1} & c_{1}

Elementary Linear Algebra 12
In Exercises 2528, confirm the identities without evaluating any of the determinants directly. \(\left|\begin{array}{ccc} a_{1}+b_{1} t & a_{2}+b_{2} t & a_{3}+b_{3} t \\ a_{1} t+b_{1} & a_{2} t+b_{2} & a_{3} t+b_{3} \\ c_{1} & c_{2} & c_{3} \end{array}\right|=\left(1-t^{2}\right)\left|\begin{arra

Elementary Linear Algebra 12
In Exercises 2528, confirm the identities without evaluating any of the determinants directly. \(\left|\begin{array}{lll}a_{1}+b_{1} & a_{1}-b_{1} & c_{1} \\ a_{2}+b_{2} & a_{2}-b_{2} & c_{2} \\ a_{3}+b_{3} & a_{3}-b_{3} & c_{3}\end{array}\right|=-2\left|\begin{array}{lll}a_{1} & b_{1} & c_{1} \

Elementary Linear Algebra 12
In Exercises 2528, confirm the identities without evaluating any of the determinants directly. \(\left|\begin{array}{lll}a_{1} & b_{1}+t a_{1} & c_{1}+r b_{1}+s a_{1} \\ a_{2} & b_{2}+t a_{2} & c_{2}+r b_{2}+s a_{2} \\ a_{3} & b_{3}+t a_{3} & c_{3}+r b_{3}+s a_{3}\end{array}\right|=\left|\begin{a

Elementary Linear Algebra 12
In Exercises 2930, show that \(det(A) = 0\) without directly evaluating the determinant. \(A=\left[\begin{array}{rrrr}-2 & 8 & 1 & 4 \\ 3 & 2 & 5 & 1 \\ 1 & 10 & 6 & 5 \\ 4 & -6 & 4 & -3\end{array}\right]\) Equation Transcription: [] Text Transcripti

Elementary Linear Algebra 12
In Exercises 2930, show that \(det(A) = 0\) without directly evaluating the determinant. \(A=\left[\begin{array}{rrrrr}-4 & 1 & 1 & 1 & 1 \\ 1 & -4 & 1 & 1 & 1 \\ 1 & 1 & -4 & 1 & 1 \\ 1 & 1 & 1 & -4 & 1 \\ 1 & 1 & 1 & 1 & -4\end{array}\right]\) Equation Transcription:

Elementary Linear Algebra 12
It can be proved that if a square matrix \(M\) is partitioned into block triangular form as \(M=\left[\begin{array}{ll}A & 0 \\C & B\end{array}\right] \quad \text { or \quadM=\left[\begin{array}{cc}A & C \\0 & B\end{array}\right]\) in which \(A\) and \(B\) are square, then \(\operatorname{det}(

Elementary Linear Algebra 12
IIt can be proved that if a square matrix \(M\) is partitioned into block triangular form as \(M=\left[\begin{array}{ll}A & 0 \\C & B\end{array}\right] \quad \text { or \quadM=\left[\begin{array}{cc}A & C \\0 & B\end{array}\right]\) in which \(A\) and \(B\) are square, then \(\operatorname{det}

Elementary Linear Algebra 12
Let \(A\) be an \(n \times n\) matrix, and let \(B\) be the matrix that results when the rows of \(A\) are written in reverse order. State a theorem that describes how \(\operatorname{det}(A)\) and \(\operatorname{det}(B)\) are related. Equation Transcription: A

Elementary Linear Algebra 12
Find the determinant of the following matrix. \(\left[\begin{array}{llll}a & b & b & b \\b & a & b & b \\b & b & a & b \\b & b & b & a \end{array}\right]\) Equation Transcription: [] Text Transcription: [a & b & b & b \\b & a & b & b \\b & b & a & b \\b & b &

Elementary Linear Algebra 12
In parts (a)(f ) determine whether the statement is true or false, and justify your answer. If \(A\) is a 4 4 matrix and \(B\) is obtained from \(A\) by interchanging the first two rows and then interchanging the last two rows, then \(det(B) = det(A)\). Equation Transcription:

Elementary Linear Algebra 12
In parts (a)(f ) determine whether the statement is true or false, and justify your answer. If \(A\) is a 3 3 matrix and \(B\) is obtained from \(A\) by multiplying the first column by 4 and multiplying the third column by \(\frac{3}{4}\(, then \(\operatorname{det}(B)=3 \operatorname{det}(A)\).

Elementary Linear Algebra 12
In parts (a)(f ) determine whether the statement is true or false, and justify your answer. If \(A\) is a 3 x 3 matrix and \(B\) is obtained from \(A\) by adding 5 times the first row to each of the second and third rows, then \(\operatorname{det}(B)=25 \operatorname{det}(A)\).

Elementary Linear Algebra 12
In parts (a)(f ) determine whether the statement is true or false, and justify your answer. If \(A\) is an \(n \times n\) matrix and \(B\) is obtained from \(A\) by multiplying each row of \(A\) by its row number, then \(\operatorname{det}(B)=\frac{n(n+1)}{2} \operatorname{det}(A)\)

Elementary Linear Algebra 12
In parts (a)(f ) determine whether the statement is true or false, and justify your answer. If \(A\) is a square matrix with two identical columns, then \(\operatorname{det}(A)=0\). Equation Transcription: Text Transcription: A det(A) = 0

Elementary Linear Algebra 12
In parts (a)(f ) determine whether the statement is true or false, and justify your answer. If the sum of the second and fourth row vectors of a 6 6 matrix \(A\) is equal to the last row vector, then \(\operatorname{det}(A)=0\). Equation Transcription: Tex

Elementary Linear Algebra 12
Find the determinant of \(A=\left[\begin{array}{rrrr}4.2 & -1.3 & 1.1 & 6.0 \\0.0 & 0.0 & -3.2 & 3.4 \\4.5 & 1.3 & 0.0 & 14.8 \\4.7 & 1.0 & 3.4 & 2.3\end{array}\right]\) by reducing the matrix to reduced row echelon form, and compare the result obtained in this way to that obtained in Exercis