Exercise · ZW-INVERSE-MATRIX-007 · ★★☆ Intermediate
Let \(A=\begin{pmatrix}1 & 2 & 0 \\ 0 & 1 & 3 \\ 0 & 0 & 1\end{pmatrix}\). What is the entry in row 1, column 3 of \(A^{-1}\)?
Choose one option.
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- C. \(6\)
Explanation
Row reducing \([A\,|\,I]\) gives \(A^{-1}=\begin{pmatrix}1 & -2 & 6\\ 0 & 1 & -3\\ 0 & 0 & 1\end{pmatrix}\). The (1,3) entry is \(6\); check by multiplying row 1 of \(A\) by column 3 of \(A^{-1}\): \(6-2\cdot3=0\).
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-004 If A and B are invertible matrices of the same size, what is (AB)^-1?
- 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-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.