Question

1. True or False. (a) For every square matrix A, if A has a zero entry, then A is not invertible. (b) If A is a square matrix such that Ax = 0 has nontrivial (nonzero) solutions, then det(A) = 0. (c) Whenever A and B are invertible n x n matrices, we have (AB)^-1 = A^-1 B^-1. (d) For all n x n matrix A, det(A^5) = det(A)^5. (e) For all square matrices A of any order, det(2A) = 2 det(A).

          1. True or False.

(a) For every square matrix A, if A has a zero entry, then A is not invertible.
(b) If A is a square matrix such that Ax = 0 has nontrivial (nonzero) solutions, then det(A) = 0.
(c) Whenever A and B are invertible n x n matrices, we have (AB)^-1 = A^-1 B^-1.
(d) For all n x n matrix A, det(A^5) = det(A)^5.
(e) For all square matrices A of any order, det(2A) = 2 det(A).

1. True or False.

(a) For every square matrix A, if A has a zero entry, then A is not invertible.
(b) If A is a square matrix such that Ax = 0 has nontrivial (nonzero) solutions, then det(A) = 0.
(c) Whenever A and B are invertible n x n matrices, we have (AB)^-1 = A^-1 B^-1.
(d) For all n x n matrix A, det(A^5) = det(A)^5.
(e) For all square matrices A of any order, det(2A) = 2 det(A).

Added by Lisa C.

Calculus: Early Transcendentals

James Stewart 8th Edition

Instant Answer

Solved by Expert Sri K

08/03/2023

Step 1

The explanation provides a counterexample of the identity matrix which is invertible despite having a zero entry. Therefore, the first statement is false. Show more…

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For every square matrix A, if A has a zero entry, then A is not invertible. If A is a matrix such that Ax = 0 has nontrivial (nonzero) solutions, then det(A) = 0. For all matrices A and B that are invertible, we have det(AB) = det(A)det(B). For all square matrices of any order, we have det(2A) = 2det(A).