Problem 18 - Algebra and Trigonometry

Question

Find the inverse of the matrix if it exists. $$ \left[\begin{array}{rrr}{5} & {7} & {4} \\ {3} & {-1} & {3} \\ {6} & {7} & {5}\end{array}\right] $$

Step-by-Step Solution

Answer

The inverse exists and is \( \begin{bmatrix} 26 & 7 & -25 \\ -3 & -5 & 3 \\ -27 & -7 & 26 \end{bmatrix} \).

Step 1: Calculate the Determinant

First, we need to check if the given matrix has an inverse, which only exists if the determinant of the matrix is non-zero. The determinant of a 3x3 matrix: \[ \begin{vmatrix} a & b & c \ d & e & f \ g & h & i \end{vmatrix} = a(ei - fh) - b(di - fg) + c(dh - eg) \]For our matrix \( A = \begin{bmatrix} 5 & 7 & 4 \ 3 & -1 & 3 \ 6 & 7 & 5 \end{bmatrix} \), substitute the values:\[ \text{det}(A) = 5((-1)(5) - (3)(7)) - 7((3)(5) - (3)(6)) + 4((3)(7) - (-1)(6)) \] Calculating: \[ = 5(-5 - 21) - 7(15 - 18) + 4(21 + 6) \] \[ = 5(-26) + 7(3) + 4(27) \] \[ = -130 + 21 + 108 \] \[ = -1 \].Since the determinant is not zero, the matrix has an inverse.

Step 2: Calculate the Matrix of Minors

Calculate each minor of the matrix, which involves the determinant of each 2x2 submatrix obtained by removing one row and one column. For example, the minor of element at (1,1) is the determinant of its complementary 2x2 matrix:For \( M_{11} \):\[ \begin{vmatrix} -1 & 3 \ 7 & 5 \end{vmatrix} = (-1)(5) - (3)(7) = -5 - 21 = -26 \].Proceed similarly for all minors:\( M_{12} = \begin{vmatrix} 3 & 3 \ 6 & 5 \end{vmatrix} = -3 \)\( M_{13} = \begin{vmatrix} 3 & -1 \ 6 & 7 \end{vmatrix} = 27 \)\( M_{21} = \begin{vmatrix} 7 & 4 \ 7 & 5 \end{vmatrix} = 7 \)\( M_{22} = \begin{vmatrix} 5 & 4 \ 6 & 5 \end{vmatrix} = 5 \)\( M_{23} = \begin{vmatrix} 5 & 7 \ 6 & 7 \end{vmatrix} = -7 \)\( M_{31} = \begin{vmatrix} 7 & 4 \ -1 & 3 \end{vmatrix} = 25 \)\( M_{32} = \begin{vmatrix} 5 & 4 \ 3 & 3 \end{vmatrix} = 3 \)\( M_{33} = \begin{vmatrix} 5 & 7 \ 3 & -1 \end{vmatrix} = -26 \).

Step 3: Calculate the Cofactor Matrix

The cofactor matrix is obtained by applying a sign change according to the positions to the matrix of minors:\[\begin{bmatrix}+M_{11} & -M_{12} & +M_{13} \-M_{21} & +M_{22} & -M_{23} \+M_{31} & -M_{32} & +M_{33}\end{bmatrix} = \begin{bmatrix}-26 & 3 & 27 \-7 & 5 & 7 \25 & -3 & -26\end{bmatrix}\].

Step 4: Transpose the Cofactor Matrix

Transpose the cofactor matrix to obtain the adjugate matrix. Transposing means swapping the rows and columns:\[\begin{bmatrix}-26 & -7 & 25 \3 & 5 & -3 \27 & 7 & -26\end{bmatrix}\].

Step 5: Multiply by the Reciprocal of the Determinant

Finally, the inverse of the matrix is obtained by multiplying the adjugate matrix by the reciprocal of the determinant (which was found to be \(-1\)). The inverse \( A^{-1} \) is:\[A^{-1} = -1 \times \begin{bmatrix}-26 & -7 & 25 \3 & 5 & -3 \27 & 7 & -26\end{bmatrix} = \begin{bmatrix}26 & 7 & -25 \-3 & -5 & 3 \-27 & -7 & 26\end{bmatrix}\].

Key Concepts

Determinant

The determinant is a special number that can be calculated from a square matrix. Imagine the determinant as a measure of a matrix's "volume" or how much space it fills. For a given 3x3 matrix, the formula involves the elements of the matrix:\[\begin{vmatrix} a & b & c \ d & e & f \ g & h & i \end{vmatrix} = a(ei - fh) - b(di - fg) + c(dh - eg)\]The determinant helps us understand whether a matrix has an inverse. If the determinant is zero, the matrix does not have an inverse. When calculating for our specific matrix, we found the determinant to be -1, which is non-zero. Hence, the matrix has an inverse.
This makes the determinant a crucial step in finding the inverse.

Matrix of Minors

A matrix of minors involves calculating a smaller matrix's determinant within the larger matrix. To find a minor for an element, remove its row and column, then calculate the determinant of the resulting 2x2 matrix. For instance, if you take the element in the first row and first column of our original matrix, the corresponding minor is computed from:\[\begin{vmatrix} -1 & 3 \ 7 & 5 \end{vmatrix}\]The operation here results in \(-26\) for the minor. Repeat this process for every element in the matrix to form the complete matrix of minors.
This step helps in preparing the data necessary to form the cofactor matrix.

Cofactor Matrix

The cofactor matrix is derived from the matrix of minors by applying alternations of signs. This is often referred to as the checkerboard pattern of pluses and minuses. For the same position of each minor matrix element, multiply the minor by \( (-1)^{i+j} \), where i and j represent the row and column indices respectively. This turns the matrix of minors into:\[\begin{bmatrix}+M_{11} & -M_{12} & +M_{13} \-M_{21} & +M_{22} & -M_{23} \+M_{31} & -M_{32} & +M_{33}\end{bmatrix} =\begin{bmatrix}-26 & 3 & 27 \-7 & 5 & 7 \25 & -3 & -26\end{bmatrix}\]This process is essential to obtaining the adjugate matrix, a step closer to finding the inverse.

Adjugate Matrix

The adjugate matrix is essentially the transpose of the cofactor matrix. Transposing a matrix means converting its rows into columns and vice versa. This transformation shifts our understanding of the matrix to be prepared for calculating the inverse. For our example, the adjugate matrix becomes:\[\begin{bmatrix}-26 & -7 & 25 \3 & 5 & -3 \27 & 7 & -26\end{bmatrix}\]From here, the final step towards the matrix inverse involves multiplying the adjugate matrix by the reciprocal of the determinant of the original matrix. This yields the inverse matrix that allows us to compute different operations, like solving systems of equations with this matrix.