Problem 19 - Algebra and Trigonometry

Question

Find the inverse of the matrix if it exists. $$ \left[\begin{array}{rrr}{1} & {2} & {3} \\ {4} & {5} & {-1} \\ {1} & {-1} & {-10}\end{array}\right] $$

Step-by-Step Solution

Answer

The inverse is \( \begin{bmatrix} 0 & -\frac{7}{4} & -\frac{1}{4} \\ -1 & \frac{19}{4} & \frac{1}{4} \\ 0 & \frac{15}{4} & \frac{1}{4} \end{bmatrix} \).

Step 1: Check if the Matrix is Invertible

To determine if a matrix has an inverse, we must first check if it is square and then compute the determinant of the matrix. An inverse exists only if the determinant is non-zero. For our 3x3 matrix, we will calculate the determinant:

Step 2: Calculate the Determinant

Use the formula for the determinant of a 3x3 matrix: \[ \text{det}(A) = a(ei-fh)-b(di-fg)+c(dh-eg) \]Substitute the matrix values: \[ \text{det}(A) = 1(5 imes (-10) - (-1) imes (-1)) - 2(4 imes (-10) - (-1) imes 1) + 3(4 imes (-1) - 5 imes 1) \]This simplifies to:\[ \text{det}(A) = 1(-50 - 1) - 2(-40 -1) + 3(-4 -5) \]\[ \text{det}(A) = -51 + 82 - 27 \]\[ \text{det}(A) = 4 \]Since the determinant is 4, which is non-zero, the inverse exists.

Step 3: Compute the Adjugate Matrix

To find the inverse, first find the cofactor matrix, then transpose it to get the adjugate matrix. This step involves calculating the minor matrices, finding their determinant, and applying the checkerboard pattern of signs.

Step 4: Find the Cofactors

Calculate the cofactors for each element. For example, the cofactor of element (1,1) is:\[ C_{11} = \text{det} \begin{bmatrix}5 & -1 \-1 & -10 \end{bmatrix} = (5)(-10) - (-1)(-1) = -51 \]Similarly, calculate other cofactors.

Step 5: Form the Adjugate Matrix

Once cofactors are calculated for all elements, the adjugate matrix is the transpose of the cofactor matrix.

Step 6: Calculate the Inverse Matrix

The inverse of a matrix is \( \frac{1}{\text{det}(A)} \times \text{adjugate matrix} \). Since \( \text{det}(A) = 4 \), multiply each element of the adjugate matrix by \( \frac{1}{4} \).

Step 7: Present the Inverse

After performing the calculations, the inverse of the matrix is found to be: \[ \begin{bmatrix} 0 & \frac{-7}{4} & \frac{-1}{4} \ -1 & \frac{19}{4} & \frac{1}{4} \ 0 & \frac{15}{4} & \frac{1}{4} \end{bmatrix} \]

Key Concepts

Determinant Calculation

To find the inverse of a matrix, the first crucial step is determining its determinant. The determinant provides important information about the matrix. It tells us if the matrix is invertible. For a 3x3 matrix, the determinant must be non-zero for the inverse to exist.

To calculate the determinant of our 3x3 matrix, we use a specific formula:

• First, label the matrix elements: \( a, b, c, d, e, f, g, h, i \).
• The formula used is: \( \text{det}(A) = a(ei-fh) - b(di-fg) + c(dh-eg) \).

Substitute the values from the matrix into this formula. Calculate the determinant by simplifying: \( \text{det}(A) = 1(-51) - 2(-41) + 3(-9) = 4 \).

As the determinant is 4, the matrix has an inverse, since it is not zero.

Adjugate Matrix

The adjugate matrix plays a vital role in finding the inverse of any square matrix. After computing the determinant, the next step is to find the adjugate, also known as the adjoint.

To create the adjugate matrix:

• Compute the cofactor matrix by calculating the minor determinate for each element.
• Apply a checkerboard pattern of signs to these minors.
• Transpose the resulting cofactor matrix, which involves swapping the rows and columns.

For our matrix, after finding all cofactors and arranging them into a matrix, take its transpose. The result is the adjugate matrix.

Cofactors

Understanding cofactors is necessary when finding the adjugate matrix. Cofactors are derived from minors and are specific to each element in the matrix.

Here's how you calculate a cofactor:

• Select an element of the matrix.
• Remove the row and column containing that element, forming a smaller matrix.
• Calculate the determinant of this smaller matrix, called a minor.
• Apply the checkerboard pattern of signs to find the cofactor: Positive sign if the sum of row and column indices is even, negative if odd.

For example, the cofactor of the element in the first row and first column in our matrix is computed from its minor's determinant and then adjusted by the sign pattern. Repeat this for all elements to build the cofactor matrix.

3x3 Matrix

The matrix in question is a 3x3 type, a standard form in math involving rows and columns filled with numbers.

This specific size, with three rows and three columns, is common because it allows calculation of properties like the determinant and the inverse using straightforward formulas.

For a 3x3 matrix, such calculations follow these outlines:

• Built as a square matrix, so each row and column has equal length.
• Includes nine elements, giving a total of nine cofactors for finding its adjugate.
• Formulas like the determinant equation are tailored to this setup, making calculations feasible since they consider all entries and intersections.

By mastering the handling of a 3x3 matrix, you can effectively approach more complex matrices with confidence.