Further Maths - Inverting Matrices
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 $MM^{-1} = M^{-1}M = I$.
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 of a two by two matrix
What’s the formula for the inverse of matrix
\[M = \left( \begin{matrix}a & b \\\\ c & d\end{matrix} \right)\]
?
What’s the formula for the inverse of a two by two matrix?
Steps for inverting a three by three matrix
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 $3 \times 3$ matrix?
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 $\frac{1}{\text{det}(M)}$
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.
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
The transpose
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