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Contents Embed Dark Mode Prev Up Next \(\require{cancel}\require{mathtools}\newcommand{\N}{\mathbb N}
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Worksheet Participate
Objectives
Calculate the inverse of a
\(2\times 2\) matrix or say why an inverse doesn’t exist.
Use the inverse and matrix multiplication to solve a 2-variable system of linear equations.
1. Let
\(A=\begin{bmatrix} 2 \amp 1 \\ 7 \amp 4 \end{bmatrix}\text{.}\)
(a) Calculate
\(A^{-1}\text{.}\)
(b) Calculate both
\(AA^{-1}\) and
\(A^{-1}A\) using matrix multiplication. What matrix do you get?
(c) What’s the size of the matrix product
\(A^{-1}\begin{bmatrix} 1 \\ 3 \end{bmatrix}\text{?}\) Is the matrix product
\(\begin{bmatrix} 1 \\ 3 \end{bmatrix}A^{-1}\) defined? Why or why not?
(d)
Use \(A^{-1}\) to solve the system
\begin{align*}
2x_1 + x_2 \amp = 1\\
7x_1+4x_2 \amp = 3
\end{align*}
(e)
Use \(A^{-1}\) to solve the system
\begin{align*}
2x_1 + x_2 \amp = 4\\
7x_1+4x_2 \amp = 14.5
\end{align*}
2. Let
\(D=\begin{bmatrix} 4 \amp -1 \\ -2 \amp 0 \end{bmatrix}\) and
\(M=\begin{bmatrix} 1 \amp -2 \\ -1 \amp 2 \end{bmatrix}\text{.}\)
(a) Calculate
\(D^{-1}\text{.}\)
(b) Say why we can’t calculate
\(M^{-1}\text{.}\)
(c) Use matrix multiplication and the matrix
\(A\) defined in
Exercise 1 to calculate both
\(AD\) and
\((AD)^{-1}\text{.}\)
(d) Use matrix multiplication and the matrix
\(A\) defined in
Exercise 1 to calculate both
\(A^{-1}D^{-1}\) and
\(D^{-1}A^{-1}\text{.}\) What do you notice?
