Exercise · ZW-INVERSE-MATRIX-005 · ★☆☆ Basic
Solve \(A\mathbf{x}=\mathbf{b}\) using the inverse, with \(A=\begin{pmatrix}1 & 2 \\ 3 & 5\end{pmatrix}\) and \(\mathbf{b}=(1,2)^T\).
Choose one option.
Show answer
- B. \((x,y)=(-1,\,1)\)
Explanation
Here \(\det A=-1\) and \(A^{-1}=\begin{pmatrix}-5 & 2\\ 3 & -1\end{pmatrix}\). Then \(\mathbf{x}=A^{-1}\mathbf{b}=(-5+4,\;3-2)=(-1,1)\). Substituting: \(-1+2=1\) and \(-3+5=2\).
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-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.