# Further Maths - Inverting Matrices

> Source: https://ollybritton.com/notes/a-level/further-maths/topics/inverting-matrices/ · Updated: 2020-09-25 · Tags: further-maths, latex-block-alt, school

### Definition of the inverse matrix
##### What is the math definition of the inverse matrix of $M$??
The inverse matrix of $M$ is the matrix $M^{-1}$ such that $MM^{-1} = M^{-1}M = I$.
^inverse-matrix-mathematical-definition
##### What's the wordy definition for an inverse matrix of $M$??
The matrix that when multiplied by the original matrix yields the identity matrix.
^inverse-matrix-wordy-definition

### Inverse of a two by two matrix
##### What's the formula for the inverse of matrix $$M = \left( \begin{matrix}a & b \\\\ c & d\end{matrix} \right)$$??
$$
\frac{1}{\text{ad - bc}} \left( \begin{matrix}d & -b \\\\ -c & a\end{matrix} \right)
$$
^two-by-two-inverse-formula-from-matrix

##### What's the formula for the inverse of a two by two matrix??
$$
\frac{1}{\text{ad - bc}} \left( \begin{matrix}d & -b \\\\ -c & a\end{matrix} \right)
$$
^two-by-two-inverse-formula

### Steps for inverting a three by three matrix
##### How many steps are there for finding the inverse of a $3 \times 3$ matrix??
5.
^three-by-three-inverse-step-count

##### What is step 1 for finding the inverse of a $3 \times 3$ matrix??
Finding the determinant.
^three-by-three-inverse-step-1

##### What is step 2 for finding the inverse of a $3 \times 3$ matrix??
Writing the matrix of minors.
^three-by-three-inverse-step-2

##### What is step 3 for finding the inverse of a $3 \times 3$ matrix??
Finding the matrix of cofactors.
^three-by-three-inverse-step-3

##### What is step 4 for finding the inverse of a $3 \times 3$ matrix??
Finding the transpose of the matrix of cofactors.
^three-by-three-inverse-step-4

##### What is step 5 for finding the inverse of a $3 \times 3$ matrix??
Multiplying the matrix of cofactors by $\frac{1}{\text{det}(M)}$
^three-by-three-inverse-step-5

### Matrix of minors, cofactors and transpose
##### What is the matrix of minors of $A$??
The matrix where each element in $A$ is replaced by its minor.
^matrix-of-minors-definition

##### What's the name for a matrix that looks like this??
PHOTO
^matrix-name-from-photo

##### What does $M$ mean when finding the inverse of a $3 \times 3$ matrix??
The matrix of minors.
^m-means-matrix-of-minors

##### What does $C$ mean when finding the inverse of a $3 \times 3$ matrix??
The matrix of cofactors.
^c-means-matrix-of-cofactors

##### What does $C^{T}$ mean when finding the inverse of a $3 \times 3$ matrix??
The transposed matrix of cofactors.
^ct-means-transposed-cofactors

##### What's a flowchart explanation for finding the inverse of a $3 \times 3$ matrix $A$??
$A \to M \to C \to C^T \to \frac{1}{\text{det}(A)} C^T$
^three-by-three-inverse-flowchart

##### If PHOTO is the matrix of minors, what is the matrix of cofactors??
PHOTO
^cofactors-from-minors-photo

### The transpose
##### What is the transpose of a matrix??
Where you "reflect" the matrix across the diagonal.
^transpose-definition

##### If the reflection line when finding the transpose of a matrix was a $y=mx$ equation, what would it be??
$y = -x$
^transpose-reflection-line

##### If PHOTO is the matrix of cofactors, what is the transposed matrix of cofactors??
PHOTO
^transposed-cofactors-from-cofactors-photo

### Properties of inverses
##### What is $AA^{-1}$??
$I$.
^a-times-a-inverse-is-identity

##### If $A^2 = I$, what does that tell you about $A$??
$$
A = A^{-1}
$$
^a-squared-identity-self-inverse

##### If $A^3 = I$, what does that tell you about $A$??
$$
A^2 = A^{-1}
$$
^a-cubed-identity-inverse

##### What's another way of writing $(AB)^{-1}$??
$$
B^{-1}A^{-1}
$$
^inverse-of-product-expanded

##### What's another way of writing $B^{-1}A^{-1}$??
$$
(AB)^{-1}
$$
^product-of-inverses-as-inverse-of-product

##### The result that $(AB)^{-1} = B^{-1}A^{-1}$ is similar to what result in vectors??
$$
\vec{AB} = b - a
$$
^inverse-product-analogous-to-vector-subtraction

---
Olly Britton — https://ollybritton.com. Machine-readable index: https://ollybritton.com/llms.txt