Unit 14.2 Matrices (Continued)
Finding the Inverse of a 2×2 Matrix
Finding the inverse of a matrix can be useful for solving systems of equations. In order for a matrix to have an inverse, it must be a square matrix
If you have the matrix M = [latex]\left[\begin{matrix}a&b\\c&d\\\end{matrix}\right][/latex]
M is a 2×2 matrix. To find the inverse of a 2×2 matrix you need to first find the determinate
Determinate of a 2×2 matrix
[latex]det={ad-bc}[/latex]
The inverse of a 2×2 matrix is 1 over the determinate times the transpose.
Inverse of a 2×2 matrix
[latex]M^{-1}=\frac{\ 1}{det}\cdot\left[\begin{matrix}d&-b\\-c&a\\\end{matrix}\right][/latex]
Finding the inverse of a 2×2 matrix
Given that [latex]M=\left[\begin{matrix}2&6\\1&4\\\end{matrix}\right][/latex]
Find M-1
First find the determinate
[latex]det=ad-bc=\left(2\ast4\right)-\left(6\ast1\right)=2[/latex]
Then find the transpose
transpose = [latex]\left[\begin{matrix}d&-b\\-c&a\\\end{matrix}\right][/latex] = [latex]\left[\begin{matrix}4&-6\\-1&2\\\end{matrix}\right][/latex]
Multiply 1/det by the transpose
[latex]M^{-1}=\frac{1}{det}\left[\begin{matrix}d&-b\\-c&a\\\end{matrix}\right]=\frac{1}{2}\left[\begin{matrix}4&-6\\-1&2\\\end{matrix}\right]=\left[\begin{matrix}2&-3\\-\frac{1}{2}&1\\\end{matrix}\right][/latex]
Finding the Inverse of a 3×3 Matrix
Finding the inverse of a 3×3 matrix involves the same steps, find the determinate and multiply it by the transpose of the matrix, but there are a few extra steps to find the transpose.
For the matrix [latex]M=\ \left[\begin{matrix}a&b&c\\d&e&f\\g&h&i\\\end{matrix}\right][/latex]
Determinate of a 3×3 matrix
[latex]det={a\cdot\left(ei-hf\right)-b\cdot\ }(di-gf)+c\cdot(dh-eg)[/latex]
Before finding the transpose, we first need to find the cofactor of the 3×3 matrix
Cofactor of a 3×3 matrix
[latex]C=\left[\begin{matrix}(ei-fh)&-(di-fg)&(dh-eg)\\-(bi-ch)&(ai-cg)&-(ah-bg)\\(bf-ce)&-(af-cd)&(ae-bd)\\\end{matrix}\right]=\left[\begin{matrix}j&k&l\\m&n&o\\p&q&r\\\end{matrix}\right][/latex]
Then we find the transpose of that matrix by swapping the rows and columns
Transpose of a 3×3 Matrix
[latex]C^T=\ \left[\begin{matrix}j&m&p\\k&n&q\\l&o&r\\\end{matrix}\right][/latex]
The inverse of a 3×3 matrix is 1 over the determinate times the transpose
Inverse of a 3×3 matrix
[latex]M^{-1}=\frac{1}{det}\cdot\ C^T[/latex]