91Ó°ÊÓ

The determinant is a special number that helps us understand certain properties of a matrix. For a 3x3 matrix like the given one, the determinant can be calculated using a specific formula. Let's denote our matrix as \(A\):
\[ A = \begin{bmatrix} 2 & 2 & -4 \ 2 & 6 & 0 \ -3 & -3 & 5 \end{bmatrix} \] The formula for the determinant of a 3x3 matrix \( A = \begin{bmatrix} a & b & c \ d & e & f \ g & h & i \end{bmatrix} \) is:
\[ \text{det}(A) = a(ei - fh) - b(di - fg) + c(dh - eg) \]
Plugging in the values from our matrix, we get:
\[ \text{det}(A) = 2(6 \times 5 - 0 \times -3) - 2(2 \times 5 - 0 \times -3) - 4(2 \times -3 - 6 \times -3) \]
Simplified, it looks like:
\[ \text{det}(A) = 2(30) - 2(10) - 4(-18) \]
Which further breaks down to: \[ \text{det}(A) = 60 - 20 + 72 = 112 \]
Since the determinant is 112 (non-zero), the matrix has an inverse.

To find the inverse of a matrix, one of the steps includes calculating the cofactor matrix. The cofactor of an element \(a_{ij}\) in a matrix is determined by deleting the row and column of that element, finding the determinant of the remaining minor matrix, and then applying a sign based on its position. This sign is \((-1)^{i+j}\).
For our matrix \(A\), we have:
Minor of \(a_{11} = \begin{vmatrix} 6 & 0 \-3 & 5 \end{vmatrix} = 6 \times 5 - 0 \times -3 = 30 \)
Similarly, we compute minors for all elements in the matrix.
After calculating minors, we convert them to cofactors by applying the appropriate sign. For example:
* \(\text{Cofactor}_{11} = (-1)^{2} \times 30 = 30\)
* \(\text{Cofactor}_{12} = (-1)^{3} \times 10 = -10\)
Once all cofactors are calculated, we form the cofactor matrix. This is an essential step towards finding the adjugate matrix.

The adjugate (or adjoint) matrix is formed by taking the transpose of the cofactor matrix. Transposing a matrix involves swapping its rows with columns.
Let's illustrate this with our 3x3 cofactor matrix:
If our cofactor matrix \(C\) is:
\[ C = \begin{bmatrix} 30 & -10 & 18 \-15 & -22 & 24 \ 0 & 6 & 10 \end{bmatrix} \]
Then the adjugate matrix, adj(A), will be:
\[ \text{adj}(A) = \begin{bmatrix} 30 & -15 & 0 \-10 & -22 & 6 \ 18 & 24 & 10 \end{bmatrix} \]
This matrix is used in the final step to find the inverse matrix.

The inverse of a matrix \(A\), denoted as \(A^{-1}\), is a matrix that, when multiplied with \(A\), yields the identity matrix. To find the inverse, we use the formula:
\[A^{-1} = \frac{1}{\text{det}(A)} \times \text{adj}(A)\]
Since we've already determined the determinant of our matrix \(A\) is 112 and we have the adjugate matrix, we can now compute the inverse:
For \(A\):
\[ \text{adj}(A) = \begin{bmatrix} 30 & -15 & 0 \-10 & -22 & 6 \ 18 & 24 & 10 \end{bmatrix} \]
The inverse matrix is computed by dividing each element of the adjugate matrix by the determinant 112:
\[ A^{-1} = \frac{1}{112} \times \begin{bmatrix} 30 & -15 & 0 \-10 & -22 & 6 \ 18 & 24 & 10 \end{bmatrix} \]
This finally gives us:
\[ A^{-1} = \begin{bmatrix} \frac{30}{112} & \frac{-15}{112} & 0 \ \frac{-10}{112} & \frac{-22}{112} & \frac{6}{112} \ \frac{18}{112} & \frac{24}{112} & \frac{10}{112} \end{bmatrix} \]
Therefore, the inverse matrix is:
\[ A^{-1} = \begin{bmatrix} \frac{15}{56} & \frac{-15}{112} & 0 \ \frac{-5}{56} & \frac{-11}{56} & \frac{3}{56} \ \frac{9}{56} & \frac{3}{14} & \frac{5}{56} \end{bmatrix} \] And that's how you find the inverse of a matrix!