 4.4.2.1: If a 4 by 4 matrix has detA = 1 2 , find det(2A), det(A), det(A 2 )...
 4.4.3.1: For these matrices, find the only nonzero term in the big formula (...
 4.4.4.1: Find the determinant and all nine cofactors Ci j of this triangular...
 4.4.1: Find the determinants of 1 1 1 1 1 1 1 2 1 1 3 1 1 4 1 1 and 2 1 0 ...
 4.4.2.2: If a 3 by 3 matrix has detA = 1, find det( 1 2 A), det(A), det(A 2 ...
 4.4.3.2: Expand those determinants in cofactors of the first row. Find the c...
 4.4.4.2: Use the cofactor matrix C to invert these symmetric matrices: A = 2...
 4.4.2: f B = M1AM, why is detB = detA? Show also that detA 1B = 1.
 4.4.2.3: Row exchange: Add row 1 of A to row 2, then subtract row 2 from row...
 4.4.3.3: True or false? (a) The determinant of S 1AS equals the determinant ...
 4.4.4.3: Find x, y, and z by Cramers Rule in equation (4): ax + by = 1 cx + ...
 4.4.3: Starting with A, multiply its first row by 3 to produce B, and subt...
 4.4.2.4: By applying row operations to produce an upper triangular U, comput...
 4.4.3.4: (a) Find the LU factorization, the pivots, and the determinant of t...
 4.4.4.4: (a) Find the determinant when a vector x replaces column j of the i...
 4.4.4: Solve 3u+2v = 7, 4u+3v = 11 by Cramers rule.
 4.4.2.5: Count row exchanges to find these determinants: det 0 0 0 1 0 0 1 0...
 4.4.3.5: Let Fn be the determinant of the 1, 1, 1 tridiagonal matrix (n by n...
 4.4.4.5: (a) Draw the triangle with vertices A = (2,2), B = (1,3), and C = (...
 4.4.5: If the entries of A and A 1 are all integers, how do you know that ...
 4.4.2.6: For each n, how many exchanges will put (row n, row n 1,..., row 1)...
 4.4.3.6: Suppose An is the n by n tridiagonal matrix with is on the three di...
 4.4.4.6: Explain in terms of volumes why det3A = 3 n detA for an n by n matr...
 4.4.6: Find all the cofactors, and the inverse or the nullspace, of " 3 5 ...
 4.4.2.7: Find the determinants of: (a) a rank one matrix A = 1 4 2 h 2 1 2i ...
 4.4.3.7: (a) Evaluate this determinant by cofactors of row 1: 4 4 4 4 1 2 0 ...
 4.4.4.7: Predict in advance, and confirm by elimination, the pivot entries o...
 4.4.7: What is the volume of the parallelepiped with four of its vertices ...
 4.4.2.8: Show how rule 6 (det = 0 if a row is zero) comes directly from rule...
 4.4.3.8: Compute the determinants of A2, A3, A4. Can you predict An? A2 = " ...
 4.4.4.8: Find all the odd permutations of the numbers {1,2,3,4}. They come f...
 4.4.8: How many terms are in the expansion of a 5 by 5 determinant, and ho...
 4.4.2.9: Suppose you do two row operations at once, going from " a b c d# to...
 4.4.3.9: How many multiplications to find an n by n determinant from (a) the...
 4.4.4.9: Suppose the permutation P takes (1,2,3,4,5) to (5,4,1,2,3). (a) Wha...
 4.4.9: If P1 is an even permutation matrix and P2 is odd, deduce from P1 +...
 4.4.2.10: If Q is an orthogonal matrix, so that Q TQ = I, prove that detQ equ...
 4.4.3.10: In a 5 by 5 matrix, does a + sign or sign go with a15a24a33a42a51 d...
 4.4.4.10: If P is an odd permutation, explain why P 2 is even but P 1 is odd.
 4.4.2.11: Prove again that detQ = 1 or 1 using only the Product rule. If det...
 4.4.3.11: If A is m by n and B is n by m, explain why det" 0 A B I # = detAB....
 4.4.4.11: Prove that if you keep multiplying A by the same permutation matrix...
 4.4.11: Explain why the point (x,y) is on the line through (2,8) and (4,7) ...
 4.4.2.12: Use row operations to verify that the 3 by 3 Vandermonde determinan...
 4.4.3.12: Suppose the matrix A is fixed, except that a11 varies from to +. Gi...
 4.4.4.12: If A is a 5 by 5 matrix with all ai j 1, then detA . Volumes or t...
 4.4.12: In analogy with the previous exercise, what is the equation for (x,...
 4.4.2.13: (a) A skewsymmetric matrix satisfies K T = K, as in K = 0 a b a 0 ...
 4.4.3.13: Compute the determinants of A, B, C from six terms. Independent row...
 4.4.4.13: Solve these linear equations by Cramers Rule x j = detBj/detA: (a) ...
 4.4.13: If the points (x,y,z), (2,1,0), and (1,1,1) lie on a plane through ...
 4.4.2.14: True or false, with reason if true and counterexample if false: (a)...
 4.4.3.14: Compute the determinants of A, B, C. Are their columns independent?...
 4.4.4.14: Use Cramers Rule to solve for y (only). Call the 3 by 3 determinant...
 4.4.14: If every row of A has either a single +1, or a single 1, or one of ...
 4.4.2.15: If every row of A adds to zero, prove that detA = 0. If every row a...
 4.4.3.15: Show that detA = 0, regardless of the five nonzeros marked by xs: A...
 4.4.4.15: Cramers Rule breaks down when detA = 0. Example (a) has no solution...
 4.4.15: If C = a b c d and D = [ u v w z], then CD = DC yields 4 equations ...
 4.4.2.16: Find these 4 by 4 determinants by Gaussian elimination: det 11 12 1...
 4.4.3.16: This problem shows in two ways that detA = 0 (the xs are any number...
 4.4.4.16: Quick proof of Cramers rule. The determinant is a linear function o...
 4.4.16: The circular shift permutes (1,2,...,n) into (2,3,...,1). What is t...
 4.4.2.17: Find the determinants of A = " 4 2 1 3# , A 1 = 1 10 " 3 2 1 4 # , ...
 4.4.3.17: Find two ways to choose nonzeros from four different rows and colum...
 4.4.4.17: If the right side b is the last column of A, solve the 3 by 3 syste...
 4.4.17: Find the determinant of A = eye(5) + ones(5) and if possible eye(n)...
 4.4.2.18: Evaluate detA by reducing the matrix to triangular form (rules 5 an...
 4.4.3.18: Place the smallest number of zeros in a 4 by 4 matrix that will gua...
 4.4.4.18: Find A 1 from the cofactor formula C T/detA. Use symmetry in part (...
 4.4.2.19: Suppose that CD = DC, and find the flaw in the following argument: ...
 4.4.3.19: (a) If a11 = a22 = a33 = 0, how many of the six terms in detA will ...
 4.4.4.19: If all the cofactors are zero, how do you know that A has no invers...
 4.4.2.20: Do these matrices have determinant 0, 1, 2, or 3? A = 0 0 1 1 0 0 0...
 4.4.3.20: How many 5 by 5 permutation matrices have detP = +1? Those are even...
 4.4.4.20: Find the cofactors of A and multiply ACT to find detA: A = 1 1 4 1 ...
 4.4.2.21: The inverse of a 2 by 2 matrix seems to have determinant = 1: detA ...
 4.4.3.21: If detA 6= 0, at least one of the n! terms in the big formula (6) i...
 4.4.4.21: Suppose detA = 1 and you know all the cofactors. How can you find A?
 4.4.2.22: Reduce A to U and find detA = product of the pivots: A = 1 1 1 1 2 ...
 4.4.3.22: Prove that 4 is the largest determinant for a 3 by 3 matrix of 1s a...
 4.4.4.22: From the formula ACT = (detA)I show that detC = (detA) n1 .
 4.4.2.23: By applying row operations to produce an upper triangular U, comput...
 4.4.3.23: How many permutations of (1,2,3,4) are even and what are they? Extr...
 4.4.4.23: (For professors only) If you know all 16 cofactors of a 4 by 4 inve...
 4.4.2.24: Use row operations to simplify and compute these determinants: det ...
 4.4.3.24: Find cofactors and then transpose. Multiply C T A and C T B by A an...
 4.4.4.24: If all entries of A are integers, and detA = 1 or 1, prove that all...
 4.4.2.25: Elimination reduces A to U. Then A = LU: A = 3 3 4 6 8 7 3 5 9 = 1 ...
 4.4.3.25: Find the cofactor matrix C and compare ACT with A 1 : A = 2 1 0 1 2...
 4.4.4.25: L is lower triangular and S is symmetric. Assume they are invertibl...
 4.4.2.26: If ai j is i times j, show that detA = 0. (Exception when A = [1].)
 4.4.3.26: The matrix Bn is the 1, 2, 1 matrix An except that b11 = 1 instead ...
 4.4.4.26: For n = 5 the matrix C contains cofactors and each 4 by 4 cofactor ...
 4.4.2.27: If ai j is i+ j, show that detA = 0. (Exception when n = 1 or 2.)
 4.4.3.27: Bn is still the same as An except for b11 = 1. So use linearity in ...
 4.4.4.27: (a) Find the area of the parallelogram with edges v = (3,2) and w =...
 4.4.2.28: Compute the determinants of these matrices by row operations: A = 0...
 4.4.3.28: The n by n determinant Cn has 1s above and below the main diagonal:...
 4.4.4.28: A box has edges from (0,0,0) to (3,1,1), (1,3,1), and (1,1,3). Find...
 4.4.2.29: What is wrong with this proof that projection matrices have detP = ...
 4.4.3.29: has 1s just above and below the main diagonal. Going down the matri...
 4.4.4.29: (a) The corners of a triangle are (2,1), (3,4), and (0,5). What is ...
 4.4.2.30: (Calculus question) Show that the partial derivatives of ln(detA) g...
 4.4.3.30: Explain why this Vandermonde determinant contains x 3 but not x 4 o...
 4.4.4.30: The parallelogram with sides (2,1) and (2,3) has the same area as t...
 4.4.2.31: (MATLAB) The Hilbert matrix hilb(n) has i, j entry equal to 1/(i + ...
 4.4.3.31: Compute the determinants S1, S2, S3 of these 1, 3, 1 tridiagonal ma...
 4.4.4.31: The Hadamard matrix H has orthogonal rows. The box is a hypercube! ...
 4.4.2.32: (MATLAB) What is a typical determinant (experimentally) of rand(n) ...
 4.4.3.32: Cofactors of those 1, 3, 1 matrices give Sn = 3Sn1 Sn2. Challenge: ...
 4.4.4.32: If the columns of a 4 by 4 matrix have lengths L1, L2, L3, L4, what...
 4.4.2.33: Using MATLAB, find the largest determinant of a 4 by 4 matrix of 0s...
 4.4.3.33: Change 3 to 2 in the upper left corner of the matrices in 32. Why d...
 4.4.4.33: Show by a picture how a rectangle with area x1y2 minus a rectangle ...
 4.4.2.34: If you know that detA = 6, what is the determinant of B? detA = row...
 4.4.3.34: With 2 by 2 blocks, you cannot always use block determinants! A B 0...
 4.4.4.34: When the edge vectors a, b, c are perpendicular, the volume of the ...
 4.4.2.35: Suppose the 4 by 4 matrix M has four equal rows all containing a, b...
 4.4.3.35: With block multiplication, A = LU has Ak = LkUk in the upper left c...
 4.4.4.35: An ndimensional cube has how many corners? How many edges? How man...
 4.4.3.36: Block elimination subtracts CA1 times the first row [A B] from the ...
 4.4.4.36: The triangle with corners (0,0), (1,0), (0,1) has area 1 2 . The py...
 4.4.3.37: A 3 by 3 determinant has three products down to the right and three...
 4.4.4.37: Polar coordinates satisfy x = r cos and y = rsin. Polar area J dr d...
 4.4.3.38: For A4 in 6, five of the 4! = 24 terms in the big formula (6) are n...
 4.4.4.38: Spherical coordinates , , give x = sin cos, y = sin sin, z = cos. F...
 4.4.3.39: For the 4 by 4 tridiagonal matrix (entries 1, 2, 1), find the five ...
 4.4.4.39: The matrix that connects r, to x, y is in 37. Invert that matrix: J...
 4.4.3.40: Find the determinant of this cyclic P by cofactors of row 1. How ma...
 4.4.4.40: The triangle with corners (0,0), (6,0), and (1,4) has area . When y...
 4.4.3.41: A=2eye(n)diag(ones(n1, 1),1)diag(ones(n1, 1),1) is the 1, 2, 1 matr...
 4.4.4.41: Let P = (1,0,1), Q = (1,1,1), and R = (2,2,1). Choose S so that PQR...
 4.4.3.42: (MATLAB) The 1, 2, 1 matrices have determinant n + 1. Compute (n + ...
 4.4.4.42: Suppose (x,y,z), (1,1,0), and (1,2,1) lie on a plane through the or...
 4.4.3.43: All Pascal matrices have determinant 1. If I subtract 1 from the n,...
 4.4.4.43: Suppose (x,y,z) is a linear combination of (2,3,1) and (1,2,3). Wha...
 4.4.4.44: If Ax = (1,0,...,0) show how Cramers Rule gives x = first column of...
 4.4.4.45: (VISA to AVIS) This takes an odd number of exchanges (IVSA, AVSI, A...
Solutions for Chapter 4: Determinants
Full solutions for Linear Algebra and Its Applications,  4th Edition
ISBN: 9780030105678
Solutions for Chapter 4: Determinants
Get Full SolutionsSince 139 problems in chapter 4: Determinants have been answered, more than 11236 students have viewed full stepbystep solutions from this chapter. This textbook survival guide was created for the textbook: Linear Algebra and Its Applications,, edition: 4. Linear Algebra and Its Applications, was written by and is associated to the ISBN: 9780030105678. This expansive textbook survival guide covers the following chapters and their solutions. Chapter 4: Determinants includes 139 full stepbystep solutions.

Additive inverse of a complex number
The opposite of a + bi, or a  bi

Arc length formula
The length of an arc in a circle of radius r intercepted by a central angle of u radians is s = r u.

Backtoback stemplot
A stemplot with leaves on either side used to compare two distributions.

Coordinate plane
See Cartesian coordinate system.

Cosine
The function y = cos x

Derivative of ƒ at x a
ƒ'(a) = lim x:a ƒ(x)  ƒ(a) x  a provided the limit exists

equation of a hyperbola
(x  h)2 a2  (y  k)2 b2 = 1 or (y  k)2 a2  (x  h)2 b2 = 1

Equivalent systems of equations
Systems of equations that have the same solution.

Graph of a function ƒ
The set of all points in the coordinate plane corresponding to the pairs (x, ƒ(x)) for x in the domain of ƒ.

Observational study
A process for gathering data from a subset of a population through current or past observations. This differs from an experiment in that no treatment is imposed.

Parametric equations
Equations of the form x = ƒ(t) and y = g(t) for all t in an interval I. The variable t is the parameter and I is the parameter interval.

Powerreducing identity
A trigonometric identity that reduces the power to which the trigonometric functions are raised.

Radicand
See Radical.

Range (in statistics)
The difference between the greatest and least values in a data set.

Real number
Any number that can be written as a decimal.

Residual
The difference y1  (ax 1 + b), where (x1, y1)is a point in a scatter plot and y = ax + b is a line that fits the set of data.

Solution of an equation or inequality
A value of the variable (or values of the variables) for which the equation or inequality is true

Solution set of an inequality
The set of all solutions of an inequality

Vector equation for a line in space
The line through P0(x 0, y0, z0) in the direction of the nonzero vector V = <a, b, c> has vector equation r = r0 + tv , where r = <x,y,z>.

Weights
See Weighted mean.