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.
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.
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 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.
