Question

Find the inverse of \\[A=\left[\begin{array}{rrr} 4 & 1 & -5 \\ -2 & 3 & 1 \\ 3 & -1 & 4 \end{array}\right]\\]

Short answer

The inverse of matrix \( A \) is \( A^{-1} = \frac{1}{98} \begin{pmatrix} 13 & 17 & 7 \\ 11 & 31 & -11 \\ -10 & 17 & 15 \end{pmatrix} \).

Step-by-step solution

Compute the Determinant

First, we need to calculate the determinant of matrix \( A \). For a 3x3 matrix \( A = \begin{pmatrix} a & b & c \ d & e & f \ g & h & i \end{pmatrix} \), the determinant is computed as: \[ \det(A) = a(ei - fh) - b(di - fg) + c(dh - eg) \] Using the elements of the matrix \( A \): \[ \det(A) = 4(3 \times 4 - 1 \times -1) - 1(-2 \times 4 - 1 \times 3) - 5(-2 \times -1 - 3 \times 3) \] Calculating each term: \( 4(12 + 1) = 52 \), \( -1(-8 - 3) = 11 \), and \( -5(2 - 9) = 35 \). Therefore, \[ \det(A) = 52 + 11 + 35 = 98 \].

Find the Matrix of Minors

The next step involves calculating the minor of each element of \( A \). The minor of an element is the determinant of the 2x2 matrix obtained by deleting the row and column of that element. Compute for each minor:1. Minor of \( a_{11} \): \( \det \begin{pmatrix} 3 & 1 \ -1 & 4 \end{pmatrix} = 3 \times 4 - (-1) \times 1 = 13 \).2. Minor of \( a_{12} \): \( \det \begin{pmatrix} -2 & 1 \ 3 & 4 \end{pmatrix} = -8 - 3 = -11 \).3. Repeat for remaining elements.

Determine the Cofactor Matrix

The cofactor is obtained by applying a sign change based on position to the minors. Use the sign pattern: \[ \begin{pmatrix} + & - & + \ - & + & - \ + & - & + \end{pmatrix} \]. Calculate:1. \( C_{11} = +13 \), \( C_{12} = -(-11) = 11 \), \( C_{13} = +(-10) \).2. Continue this pattern for each element of the matrix to get the full cofactor matrix.

Transpose the Cofactor Matrix

Create the adjugate matrix (transpose of the cofactor matrix). If the cofactor matrix is \[ C = \begin{pmatrix} 13 & 11 & -10 \ 17 & 31 & 17 \ 7 & -11 & 15 \end{pmatrix} \], then its transpose is \[ C^T = \begin{pmatrix} 13 & 17 & 7 \ 11 & 31 & -11 \ -10 & 17 & 15 \end{pmatrix} \].

Divide by the Determinant

Finally, divide each element of the transpose of the cofactor matrix by the determinant of \( A \), which we calculated earlier as 98. This gives the inverse matrix \( A^{-1} \):\[ A^{-1} = \frac{1}{98} \begin{pmatrix} 13 & 17 & 7 \ 11 & 31 & -11 \ -10 & 17 & 15 \end{pmatrix} \]

Key concepts

Determinant Calculation

Calculating the determinant of a square matrix is essential for finding its inverse. Specifically, a matrix can only have an inverse if its determinant is non-zero.

For a 3x3 matrix, the determinant is a unique value obtained from a specific arithmetic formula. Consider a matrix \( A = \begin{pmatrix} a & b & c \ d & e & f \ g & h & i \end{pmatrix} \). The determinant is computed through this formula:
\[ \det(A) = a(ei - fh) - b(di - fg) + c(dh - eg) \]
This might look complex, but remember it's about breaking it down into simpler parts. Each term—\( a(ei - fh) \), \( b(di - fg) \), \( c(dh - eg) \)—includes the subtraction of the products of the elements within the matrix.
For our matrix \( A \), performing this calculation provides a determinant of \( 98 \), indicating the matrix is invertible.

Cofactor Matrix

After computing the determinant, the next step is to derive the cofactor matrix, which is vital in determining the inverse of a matrix.

A cofactor is derived from a minor. The minor of an element in a matrix is the determinant of the smaller 2x2 matrix that remains after removing its corresponding row and column. Once the matrix of minors is obtained, each minor is adjusted with a sign based on its position to form the cofactor. This sign is determined by the checkerboard pattern of plus and minus:

• ‘+’ for positions where (row + column) is even,
• and ‘-’ for positions where (row + column) is odd.

Applying this pattern allows us to form the cofactor matrix. For example, if the element at position \( (1,1) \) is \( + \) and its minor is \( 13 \), the cofactor will also be \( 13 \). Continue this for all elements to complete the matrix.

Adjugate Matrix

The adjugate, or adjoint matrix, is crucial for obtaining the inverse, and it is derived by transposing the cofactor matrix.

Let's delve a bit into what transposing means. To transpose a matrix, you simply switch its rows with its columns.
If your original cofactor matrix is
\[ C = \begin{pmatrix} 13 & 11 & -10 \ 17 & 31 & 17 \ 7 & -11 & 15 \end{pmatrix} \]
Then the adjugate, \( C^T \), becomes:
\[ C^T = \begin{pmatrix} 13 & 17 & 7 \ 11 & 31 & -11 \ -10 & 17 & 15 \end{pmatrix} \]
This transformation is integral since multiplying the adjugate by 1 over the determinant gives you the inverse. Remember, without the determinant indicator, these steps cannot achieve our desired inverse.

Matrix of Minors

The matrix of minors is a fundamental step toward finding the cofactor matrix and, ultimately, the inverse matrix.
Each element of the matrix of minors is the determinant of the small 2x2 matrix obtained by removing the corresponding row and column of a given element in the original matrix.

• For the element at position (1,1), remove the first row and column and compute the determinant of this smaller 2x2 matrix.
• Continue this process for each element of the matrix to build the entire matrix of minors.

Let's say you compute
\( \det\begin{pmatrix} 3 & 1 \ -1 & 4 \end{pmatrix} = 13 \), which becomes part of the matrix of minors. Thus, you follow this pattern for all elements.
Building this sets a foundational block closest to achieving the matrix inverse and provides a better understanding of the matrix's structure.