Unit 14.2 Matrices (Continued)

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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 = $[\begin{matrix} a & b \\ c & d \end{matrix}]$

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

$d e t = a d - b c$

The inverse of a 2×2 matrix is 1 over the determinate times the transpose.

Inverse of a 2×2 matrix

$M^{- 1} = \frac{1}{d e t} \cdot [\begin{matrix} d & - b \\ - c & a \end{matrix}]$

Finding the inverse of a 2×2 matrix

Given that $M = [\begin{matrix} 2 & 6 \\ 1 & 4 \end{matrix}]$

Find M^-1

First find the determinate

$d e t = a d - b c = (2 * 4) - (6 * 1) = 2$

Then find the transpose

transpose = $[\begin{matrix} d & - b \\ - c & a \end{matrix}]$ = $[\begin{matrix} 4 & - 6 \\ - 1 & 2 \end{matrix}]$

Multiply 1/det by the transpose

$M^{- 1} = \frac{1}{d e t} [\begin{matrix} d & - b \\ - c & a \end{matrix}] = \frac{1}{2} [\begin{matrix} 4 & - 6 \\ - 1 & 2 \end{matrix}] = [\begin{matrix} 2 & - 3 \\ - \frac{1}{2} & 1 \end{matrix}]$

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 $M = [\begin{matrix} a & b & c \\ d & e & f \\ g & h & i \end{matrix}]$

Determinate of a 3×3 matrix

$d e t = a \cdot (e i - h f) - b \cdot (d i - g f) + c \cdot (d h - e g)$

Before finding the transpose, we first need to find the cofactor of the 3×3 matrix

Cofactor of a 3×3 matrix

$C = [\begin{matrix} (e i - f h) & - (d i - f g) & (d h - e g) \\ - (b i - c h) & (a i - c g) & - (a h - b g) \\ (b f - c e) & - (a f - c d) & (a e - b d) \end{matrix}] = [\begin{matrix} j & k & l \\ m & n & o \\ p & q & r \end{matrix}]$

Then we find the transpose of that matrix by swapping the rows and columns

Transpose of a 3×3 Matrix

$C^{T} = [\begin{matrix} j & m & p \\ k & n & q \\ l & o & r \end{matrix}]$

The inverse of a 3×3 matrix is 1 over the determinate times the transpose

Inverse of a 3×3 matrix

$M^{- 1} = \frac{1}{d e t} \cdot C^{T}$

Unit 14.2 Matrices (Continued)

Finding the Inverse of a 2×2 Matrix

Finding the Inverse of a 3×3 Matrix

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