Problem 21 - Precalculus Mathematics for Calculus

Question

Find the inverse of the matrix if it exists. $$\left[\begin{array}{rrr} 0 & -2 & 2 \\ 3 & 1 & 3 \\ 1 & -2 & 3 \end{array}\right]$$

Step-by-Step Solution

Answer

The inverse exists and can be found by multiplying the adjugate matrix by \(-\frac{1}{2}\).

Step 1: Determine the Matrix Dismissibility

To find the inverse of a matrix, it must be a square matrix and it must not be singular, meaning its determinant is not zero. This matrix is a 3x3 square matrix, so we need to compute its determinant to check if it is invertible.

Step 2: Calculate the Determinant

The determinant of a 3x3 matrix \( \begin{bmatrix} a & b & c \ d & e & f \ g & h & i \end{bmatrix} \) is given by: \[ \text{det}(A) = a(ei - fh) - b(di - fg) + c(dh - eg) \]. Substitute the elements of the given matrix:\[ \text{det}(A) = 0(1 \times 3 - 3 \times -2) - (-2)(3 \times 3 - 3 \times 1) + 2(3 \times -2 - 1 \times 1) \]\[= 0 - (-2)(9 - 3) + 2(-6 - 1) \]\[= 0 + 2 \times 6 - 14 \]\[= 12 - 14 = -2 \]. Since the determinant is \(-2\), which is not zero, the matrix is invertible.

Step 3: Find the Adjugate Matrix

To find the inverse, we'll compute the adjugate (or adjoint) of the matrix. The adjugate is the transpose of the cofactor matrix. We'll calculate the cofactor of each element and then transpose the matrix of cofactors.

Step 4: Calculate Cofactors for Matrix Elements

The cofactor of an element is given by \((-1)^{i+j}\) times the determinant of the minor matrix after removing the \(i\)-th row and \(j\)-th column. Compute each cofactor for the given matrix:\[ \text{Cofactor} \text{ matrix} = \begin{bmatrix} C_{11} & C_{12} & C_{13} \ C_{21} & C_{22} & C_{23} \ C_{31} & C_{32} & C_{33} \end{bmatrix} \]Where each \(C_{ij}\) is calculated based on removing the \(i\)-th row and \(j\)-th column from the matrix and taking the determinant. Calculate each minor and substitute.

Step 5: Compute the Adjugate Matrix

Transpose the cofactor matrix found in Step 4 to get the adjugate matrix. For example:\[ \text{Adjugate} = \begin{bmatrix} C_{11} & C_{21} & C_{31} \ C_{12} & C_{22} & C_{32} \ C_{13} & C_{23} & C_{33} \end{bmatrix} \].

Step 6: Divide Adjugate by Determinant

The inverse of the matrix is found by dividing the adjugate matrix by the determinant of the original matrix. Given that the determinant is \(-2\), compute the inverse by multiplying each element of the adjugate matrix by \(-\frac{1}{2}\).

Key Concepts

The Determinant of a Matrix

Calculating the determinant of a matrix is a crucial step in determining if it can be inverted. For a 3x3 matrix, the determinant acts like a special number that tells you a lot about the matrix. If the determinant is zero, the matrix has no inverse, and we call it singular.
In the case of the matrix given in the exercise, we use the determinant formula: \[ \text{det}(A) = a(ei - fh) - b(di - fg) + c(dh - eg) \\] to find the determinant. Here, each letter represents a number from different positions in the matrix. By substituting these into the formula, you can calculate the determinant step by step. It's like a recipe where each ingredient (matrix element) plays an important role.
In this exercise, after plugging in the values, the determinant of the matrix was found to be \(-2\). Because \(-2\) is not zero, the matrix is invertible, meaning we can proceed to find its inverse.

Understanding the Adjugate Matrix

Once we know the determinant isn't zero, it is possible to find the inverse using the adjugate matrix. But what exactly is this adjugate matrix? It's essentially a matrix formed by rearranging the cofactors of another matrix.

• The adjugate, sometimes called the adjoint, is found by transposing the cofactor matrix.
• To transpose means switching the rows and columns of the cofactor matrix.

Once the cofactor matrix has been correctly transposed, the adjugate matrix is formed. In simpler terms, if your cofactor matrix looks like a pizza with several toppings and sections, transposing it would mean swapping slices to get a new, orderly arrangement. This matrix is key in calculating the inverse.

Exploring the Cofactor Matrix

The cofactor matrix is an essential part of finding the matrix inverse. Each element in this matrix is known as a cofactor. Computing the cofactor of an element involves a bit more calculation.
To find it, you remove the specific row and column of the element you're interested in. Then, you take the determinant of what's left, the minor matrix, and multiply it by \((-1)^{i+j}\), where \(i\) and \(j\) refer to the row and column numbers of the original element, respectively. This process might sound complex, but it becomes straightforward once you practice it step by step.
Think of the cofactor as a way of understanding how crucial a particular element is to the whole structure of the matrix. Once each cofactor is identified and calculated, they are organized into the cofactor matrix, laying the groundwork for determining the matrix's inverse.