Problem 17 - Precalculus with Limits

Question

In Exercises 13-24, find the inverse of the matrix (if it exists). \(\left[ \begin{array}{r} 3 && 1 \\ 4 && 2\end{array} \right]\)

Step-by-Step Solution

Answer

The inverse of the given matrix is: \( \left[ \begin{array}{r} 1 && - \frac{1}{2} \ -2 && \frac{3}{2}\end{array} \right]\)

Step 1: Calculate determinant

Calculate the determinant of the given matrix, denoted as \( det(A)\). For a 2x2 matrix \( \left[ \begin{array}{r} a && b \ c && d\end{array} \right]\), the determinant is calculated as \(ad - bc\). Therefore, for the given matrix, \( det(A) = 3*2 - 1*4 = 6 - 4 = 2.\)

Step 2: Determine if matrix is invertible

Check if the determinant of the matrix is zero or not. If the determinant is zero, it means the matrix is singular and it does not have an inverse. If the determinant is non-zero, it means the matrix is non-singular and it has an inverse. In this case, \( det(A) = 2 \) which is non-zero, so the matrix has an inverse.

Step 3: Calculate adjugate matrix

Calculate the adjugate of the given matrix. For a 2x2 matrix \( \left[ \begin{array}{r} a && b \ c && d\end{array} \right]\), the adjugate matrix is \( \left[ \begin{array}{r} d && -b \ -c && a\end{array} \right]\). Therefore, for the given matrix, the adjugate matrix is \( \left[ \begin{array}{r} 2 && -1 \ -4 && 3\end{array} \right]\).

Step 4: Calculate the inverse of the matrix

Obtain the inverse of the matrix by multiplying the scalar reciprocal of the determinant with the adjugate matrix. The reciprocal of the determinant \(2\) is \( \frac{1}{2}\). Therefore, matrix inverse A^-1 is \( \frac{1}{2} \) times \( \left[ \begin{array}{r} 2 && -1 \ -4 && 3\end{array} \right]\) which gives \( \left[ \begin{array}{r} 1 && - \frac{1}{2} \ -2 && \frac{3}{2}\end{array} \right]\).

Key Concepts

Determinant

The determinant is a special number associated with square matrices. For a 2x2 matrix, it is computed as the difference between the product of the diagonal elements. Let's take a look the matrix:
\[ \begin{bmatrix} 3 & 1 \ 4 & 2 \end{bmatrix} \]
The formula for the determinant of a 2x2 matrix \( \begin{bmatrix} a & b \ c & d \end{bmatrix} \) is \( ad - bc \). In this example, it becomes \( 3 \times 2 - 1 \times 4 = 6 - 4 \), resulting in a determinant of 2.
A few important things to remember about determinants:

• If the determinant is zero, the matrix is singular and does not have an inverse.
• If the determinant is non-zero, the matrix is non-singular and may have an inverse.

In this matrix, the determinant is 2, meaning it's not zero, so the matrix could be invertible.

Adjugate Matrix

The adjugate matrix is crucial for finding the inverse of a matrix. An adjugate (or adjoint) is essentially a matrix of cofactors transposed. For 2x2 matrices, the process is quite straightforward.
Given the matrix:\[ \begin{bmatrix} 3 & 1 \ 4 & 2 \end{bmatrix} \]
The adjugate matrix can be found by swapping the positions of the elements in the main diagonal and changing the signs of the other two elements. This transforms the matrix into:

• Swap the diagonal elements (3 and 2 become 2 and 3)
• Change signs of off-diagonals (1 becomes -1, and 4 becomes -4)
• Thus, it becomes:

\[ \begin{bmatrix} 2 & -1 \ -4 & 3 \end{bmatrix} \]
The adjugate is instrumental in calculating the matrix inverse.

Invertible Matrix

An invertible matrix, also known as a non-singular matrix, is one that has an inverse. To determine if a matrix is invertible, we check its determinant.
For the given matrix:

• If \( \text{det}(A) eq 0 \), the matrix has an inverse.
• If \( \text{det}(A) = 0 \), the matrix does not have an inverse and is termed singular.

The determinant of our matrix is 2, which is not zero. This confirms that the matrix is invertible.
The inverse of a 2x2 matrix \( A \) is calculated as the reciprocal of the determinant multiplied by the adjugate matrix:

• Step 1: Find determinant (we did: 2).
• Step 2: Compute adjugate matrix (we did: \( \begin{bmatrix} 2 & -1 \ -4 & 3 \end{bmatrix} \)).
• Step 3: Multiply adjugate by \( \frac{1}{\text{det}(A)} \).

Therefore, the inverse matrix computed is:\[ \frac{1}{2} \begin{bmatrix} 2 & -1 \ -4 & 3 \end{bmatrix} = \begin{bmatrix} 1 & -\frac{1}{2} \ -2 & \frac{3}{2} \end{bmatrix} \]

Non-Singular Matrix

The terms non-singular matrix and invertible matrix are often used interchangeably. In essence, a non-singular matrix is one with a non-zero determinant.
Let's reiterate its key points:

• A non-zero determinant indicates non-singular (invertible).
• If \( det(A) = 0 \), it's singular (not invertible).

Our matrix, with a determinant of 2, meets the criteria for being non-singular.
Why is this important? Non-singular matrices:

• Can be "undone" or "reversed" using their inverses.
• Play vital roles in solving systems of linear equations.
• Are central in linear transformations and more advanced mathematics.

For practical calculations, always check the determinant first!