Further Maths - Inverting Matricies

Created: September 25, 2020

What is the math definition of the inverse matrix of $M$?

The inverse matrix of $M$ is the matrix $M^{-1}$ such that $MM^{-1} = M^{-1}M = I$.

What’s the wordy definition for an inverse matrix of $M$?

The matrix that when multiplied by the original matrix yields the identity matrix.

What’s the formula for the inverse of matrix

\[M = \left( \begin{matrix}a & b \\\\ c & d\end{matrix} \right)\]

?
\[\frac{1}{\text{ad - bc} M} \left( \begin{matrix}d & -b \\\\ -c & a\end{matrix} \right)\]

What’s the formula for the inverse of a two by two matrix?
\[\frac{1}{\text{ad - bc} M} \left( \begin{matrix}d & -b \\\\ -c & a\end{matrix} \right)\]

How many steps are there for finding the inverse of a $3 \times 3$ matrix?

5.

What is step 1 for finding the inverse of a $3 \times 3$ matix?

Finding the determinant.

What is step 2 for finding the inverse of a $3 \times 3$ matrix?

Writing the matrix of minors.

What is step 3 for finding the inverse of a $3 \times 3$ matrix?

Finding the matrix of cofactors.

What is step 4 for finding the inverse of a $3 \times 3$ matrix?

Finding the transpose of the matrix of cofactors.

What is step 5 for finding the inverse of a $3 \times 3$ matrix?

Multiplying the matrix of cofactors by $\frac{1}{\text{det}(M)}$

What is the matrix of minors of $A$?

The matrix where each element in $A$ is replaced by its minor.

What’s the name for a matrix that looks like this?

PHOTO

What does $M$ mean when finding the inverse of a $3 \times 3$ matrix?

The matrix of minors.

What does $C$ mean when finding the inverse of a $3 \times 3$ matrix?

The matrix of cofactors.

What does $C^{T}$ mean when finding the inverse of a $3 \times 3$ matrix?

The transposed matrix of cofactors.

What’s a flowchart explanation for finding the inverse of a $3 \times 3$ matrix $A$?

$A \to M \to C \to C^T \to \frac{1}{\text{det}(A)} C^T$

If PHOTO is the matrix of minors, what is the matrix of cofactors?

PHOTO

What is the transpose of a matrix?

Where you “reflect” the matrix across the diagonal.

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

$y = -x$

If PHOTO is the matrix of cofactors, what is the transposed matrix of cofactors?

PHOTO

What is $AA^{-1}$?

$I$.

If $A^2 = I$, what does that tell you about $A$?
\[A = A^{-1}\]
If $A^3 = I$, what does that tell you about $A$?
\[A^2 = A^{-1}\]

What’s another way of writing $(AB)^{-1}$?
\[B^{-1}A^{-1}\]

What’s another way of writing $B^{-1}A^{-1}$?
\[(AB)^{-1}\]

The result that $(AB)^{-1} = B^{-1}A^{-1}$ is similar to what result in vectors?
\[\vec{AB} = b - a\]


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