Problem 16 - Precalculus: Mathematical for Calculus

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 calculated using the adjoint method.

Step 1: Find the Determinant of the Matrix

To determine if the inverse exists, first we calculate the determinant of the matrix \( A = \begin{bmatrix} 5 & 7 & 4 \ 3 & -1 & 3 \ 6 & 7 & 5 \end{bmatrix} \). We use the formula: \( \text{det}(A) = a(ei-fh) - b(di-fg) + c(dh-eg) \). Substitute in the values: \( 5((-1)(5)-7(3)) - 7(3*5-3*6) + 4(3*7 - (-1)6) \). Calculate each term: \(-5(5 + 21) - 7(15 - 18) + 4(21 + 6) \). This simplifies to \(-5(26) + 7(3) + 4(27) \). Thus, \( \text{det}(A) = -130 + 21 + 108 = -1 \).

Step 2: Check if the Determinant is Zero

An inverse for a matrix exists only if the determinant is non-zero. In this case, the determinant of the matrix is \(-1\), which is not zero. Therefore, the matrix is invertible.

Step 3: Compute the Inverse Using the Adjoint Method

The inverse of a matrix \( A \) is given by \( A^{-1} = \frac{1}{\text{det}(A)} \cdot \text{adj}(A) \). First, find the cofactor matrix of \( A \) and then transpose it to get the adjoint. The cofactor for each element can be found by removing the row and column of that element and calculating the determinant of the resulting 2x2 matrix.

Step 4: Calculate the Cofactor Matrix

For element 5: remove row 1 and col 1, the minor is \( \begin{bmatrix} -1 & 3 \ 7 & 5 \end{bmatrix} \) with determinant \(-1 \cdot 5 - 3 \cdot 7 = -5 - 21 = -26 \). For element 7: \( \begin{bmatrix} 3 & 3 \ 6 & 5 \end{bmatrix} \) determinant \(3 \cdot 5 - 3 \cdot 6 = 15 - 18 = -3 \), sign is negative, so cofactor is 3. Continue this process for every element.

Step 5: Find the Adjoint Matrix

Transpose the cofactor matrix to form the adjoint matrix. E.g., if the cofactor matrix is \( \begin{bmatrix} -26 & 3 & y \ x & z & t \ u & v & w \end{bmatrix} \), then the adjoint is \( \begin{bmatrix} -26 & x & u \ 3 & z & v \ y & t & w \end{bmatrix} \). Calculate all needed cofactors to fill this matrix.

Step 6: Multiply the Adjoint by the Reciprocal of the Determinant

Finally, to find the inverse \( A^{-1} \), multiply each element of the adjoint matrix by \( \frac{1}{-1} \), essentially negating each element of the adjoint matrix.

Key Concepts

Determinant

The determinant is a crucial concept when exploring the inverse of a matrix. Think of it like a special number that can tell you if a matrix is invertible. To find the determinant of a 3x3 matrix like \( A \), you use the rule of Sarrus, which involves multiplying and subtracting values from the matrix. Consider the matrix \( A = \begin{bmatrix} 5 & 7 & 4 \ 3 & -1 & 3 \ 6 & 7 & 5 \end{bmatrix} \). To find its determinant, we apply the formula:

• \[ \text{det}(A) = a(ei - fh) - b(di - fg) + c(dh - eg) \]
• By substituting in the values, you calculate products and differences: \( 5((-1)(5) - 7(3)) - 7(3 \times 5 - 3 \times 6) + 4(3 \times 7 - (-1) \times 6) \).
• Carrying out these calculations step by step gives the determinant as \(-1\).

The determinant is non-zero, specifically \(-1\), thus making the matrix invertible.

Adjoint Matrix

The adjoint of a matrix is an essential matrix used to calculate the inverse. We arrive at the adjoint matrix by computing the cofactor matrix and then transposing it. The cofactor matrix is a collection of determinant values obtained from removing specific rows and columns from the original matrix. It sounds complex but can be broken down:

• Compute the cofactor for each element by eliminating its row and column, which gives a smaller 2x2 matrix.
• Each of these smaller matrices has its own determinant, forming part of the cofactor matrix.

Once the cofactor matrix is found, you transpose it by swapping rows with columns to form the adjoint. This matrix plays a critical part because the inverse is computed as \( A^{-1} = \frac{1}{\text{det}(A)} \times \text{adj}(A) \).

Cofactor Matrix

Understanding the cofactor matrix involves looking at parts of the original matrix individually. A cofactor is a signed minor. To get these minors for a 3x3 matrix like \( A \):

• For each element of the matrix, eliminate that element's row and column, leaving a 2x2 minor matrix.
• Calculate the determinant of each minor, with the sign determined by \((-1)^{i+j}\), where \(i\) and \(j\) are the row and column indices.

For instance, the cofactor corresponding to the element \(5\) in matrix \( A \) is derived from \[ \begin{bmatrix} -1 & 3 \ 7 & 5 \end{bmatrix} \], evaluating to \(-26\). This building of cofactors helps create the cofactor matrix, an intermediate step toward computing the adjoint.

Invertible Matrix

An invertible matrix, also known as a non-singular matrix, is one that has an inverse. The existence of an inverse is crucial because it means there is another matrix that can "undo" the initial matrix through multiplication.

• If a matrix's determinant is zero, it is not invertible. The presence of a non-zero determinant, like \(-1\) in our example, assures invertibility.
• The inverse is used in myriad applications such as solving systems of equations, transforming coordinate systems, and more.
• To find the inverse of an invertible matrix \( A \), use the formula: \( A^{-1} = \frac{1}{\text{det}(A)} \cdot \text{adj}(A) \), which scales each element of the adjoint matrix by \( \frac{1}{\text{det}(A)} \).

The matrix \( A \) in the example is indeed invertible, as its determinant is -1.