Exercise · ZW-INVERSE-MATRIX-004 · ★☆☆ Basic
If \(A\) and \(B\) are invertible matrices of the same size, what is \((AB)^{-1}\)?
Choose one option.
Show answer
- D. \(B^{-1}A^{-1}\)
Explanation
The order reverses: \((AB)(B^{-1}A^{-1})=A(BB^{-1})A^{-1}=AA^{-1}=I\). In general \(A^{-1}B^{-1}\) fails because matrix multiplication is not commutative.
Review the concept: Inverse matrix
The inverse of a square matrix \(A\) is the matrix \(A^{-1}\) that gives the identity \(I\) when multiplied by \(A\) on either side; if it exists, \(A\) is called invertible.
\[AA^{-1}=A^{-1}A=I\]
More practice problems: inverse matrix
- ZW-INVERSE-MATRIX-001 Compute the inverse of [2 1; 5 3].
- ZW-INVERSE-MATRIX-002 Which of these matrices has NO inverse?
- ZW-INVERSE-MATRIX-003 Compute the inverse of the diagonal matrix [2 0 0; 0 -4 0; 0 0 5].
- ZW-INVERSE-MATRIX-005 Solve Ax=b using the inverse, with A=[1 2; 3 5] and b=(1,2)^T.
- ZW-INVERSE-MATRIX-006 A square matrix A satisfies A^2-3A+I=0. Which expression is A^-1?
- ZW-INVERSE-MATRIX-007 Let A=[1 2 0; 0 1 3; 0 0 1]. What is the entry in row 1, column 3 of A^-1?
- ZW-INVERSE-MATRIX-008 If A is an invertible 2×2 matrix, what is (3A)^-1?
- ZW-INVERSE-MATRIX-009 Select all values of k for which [k 2; 8 k] is NOT invertible.