Determine whether each statement is true or false. If false, give a counterexample. a. Interchanging 2...

Determine whether each statement is true or false. If false, give a counterexample.

a. Interchanging 2 rows of a given matrix changes the sign of its determinant.

b. If A is a square matrix, then the cofactor Cij of the entry aij is the determinant of the matrix obtained by deleting the ith row and jth column of A.

c. Every nonsingular matrix can be written as the product of elementary matrices.

d. If A is invertible, the AX = 0 has only the trivial solution.

e. If A and B are invertible matrices of order x, then AB is invertible and (AB)^-1 = (A^-1)(B^-1).

f. If A and B are matrices such that AB is defined, then (AB)^T = (A^T) (B^T).

g. Matrix A is symmetric if A = A^T.

h. Matrix multiplication is associative.

i. Matrix multiplication is commutative.

j. Every matrix has an additive inverse.

k. Every homogeneous system of linear equations is consistent.

l. A system of 2 linear equations in three variables is always consistent.

m. A linear system can have exactly two solutions.

Expert Solution

Colby Messinger answered 1 month ago

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