Inverse Matrix Properties

A product of invertible n×n matrices is​ invertible, and the inverse of the product is the product of their inverses in the same order.

A. ​False; if A and B are invertible​ matrices, then (AB)^−1 = B^−1 * A^−1.

Suppose A is n×n and the equation Ax=b has a solution for each b in ℝ^n. Explain why A must be invertible.​ [Hint: Is A row equivalent to Isubscriptn​?]

An n×n matrix A is invertible if and only if A is row equivalent to I subscript n​, and any sequence of elementary row operations that reduces A to I subscript n also transforms I subscript n into A^−1. A pivot position in a matrix A is a location in A that corresponds to a leading 1 in the reduced echelon form of A. A pivot column is a column of A that contains a pivot position. Recall that two matrices are row equivalent if one can be changed to the other by a sequence of elementary row operations. If the equation Ax=b has a solution for each b in ℝ^n​, then A has a pivot position in each row. Since A is​ square, the pivots must be on the diagonal of A. It follows that A is row equivalent to I subscript n. ​Therefore, A is invertible.

If A can be row reduced to the identity​ matrix, then A must be invertible.

True; since A can be row reduced to the identity​ matrix, A is row equivalent to the identity matrix. Since every matrix that is row equivalent to the identity is​ invertible, A is invertible.

If A is an invertible n×n matrix, then the equation Ax=b is consistent for each b in ℝ^n.

True; since A is​ invertible, A^−1*b exists for all b in ℝ^n. Define x = A^−1*b. Then Ax=b.

If A is​ invertible, then the inverse of A^−1 is A itself.

True; since A^−1 is the inverse of​ A, A^−1 * A = I = A*A^−1. Since A^−1*A = I = A*A^−1​, A is the inverse of A^−1.

Each elementary matrix is invertible.

True; since each elementary matrix corresponds to a row​ operation, and every row operation is​ reversible, every elementary matrix has an inverse matrix.

If A and B are n×n and​ invertible, then A^−1*B^−1 is the inverse of AB.

​False; B^−1*A^−1 is the inverse of AB.

If A is​ invertible, then elementary row operations that reduce A to the identity I subscript n also reduce A^−1 to I subscript n.

​False; if A is​ invertible, then the row operations required to reduce A to the identity correspond to some product of elementary matrices E sub1 * Esub2 * Esub3 ••• Esubp. Then the row operations required to reduce A^−1 to the identity would correspond to the product Esubp^−1 ••• Esub3^−1 * Esub2^−1 * Esub1^−1.

If A = [a b c d] and ab−cd ≠ ​0, then A is invertible.

​False; if ad−bc ≠ ​0, then A is invertible.

In order for a matrix B to be the inverse of​ A, both equations AB=I and BA=I must be true.

​True, by definition of invertible.

If A=[a b c d] and ad=​bc, then A is not invertible.

​True; if ad=bc then ad−bc=​0, and 1/(ad−bc)*[d −b −c a] is undefined.