Problem 18 - College Algebra with Modeling and Visualization

Question

( Refer to Examples 3-5.) LetA be the given matrix. Find \(A^{-1}\) without a calculator. $$ \left[\begin{array}{ll} 1 & 3 \\ 2 & 5 \end{array}\right] $$

Step-by-Step Solution

Answer

The inverse of the matrix \( \begin{bmatrix} 1 & 3 \\ 2 & 5 \end{bmatrix} \) is \( \begin{bmatrix} -5 & 3 \\ 2 & -1 \end{bmatrix} \).

Step 1: Determine the matrix size

The given matrix \( A \) is a 2x2 matrix, which is necessary to find its inverse using the determinant method.

Step 2: Find the determinant

Calculate the determinant of matrix \( A \) using the formula for a 2x2 matrix: \[ \text{det}(A) = ad - bc \] For \( A = \begin{bmatrix} 1 & 3 \ 2 & 5 \end{bmatrix} \), \[ \text{det}(A) = (1)(5) - (3)(2) = 5 - 6 = -1 \]

Step 3: Confirm if the matrix is invertible

Since the determinant is \(-1\) (non-zero), matrix \( A \) is invertible. We can proceed to find the inverse.

Step 4: Apply the formula for the inverse of a 2x2 matrix

For a generic 2x2 matrix \( \begin{bmatrix} a & b \ c & d \end{bmatrix} \), the inverse is given by \[ A^{-1} = \frac{1}{\text{det}(A)} \begin{bmatrix} d & -b \ -c & a \end{bmatrix} \] Substitute the values from matrix \( A \) to find \( A^{-1} \):\[ A^{-1} = \frac{1}{-1} \begin{bmatrix} 5 & -3 \ -2 & 1 \end{bmatrix} = \begin{bmatrix} -5 & 3 \ 2 & -1 \end{bmatrix} \]

Step 5: Verify the inverse

Multiply \( A \) by its calculated inverse \( A^{-1} \) to verify if the result is the identity matrix \( I \): \[ A \cdot A^{-1} = \begin{bmatrix} 1 & 3 \ 2 & 5 \end{bmatrix} \begin{bmatrix} -5 & 3 \ 2 & -1 \end{bmatrix} = \begin{bmatrix} (1)(-5) + (3)(2) & (1)(3) + (3)(-1) \ (2)(-5) + (5)(2) & (2)(3) + (5)(-1) \end{bmatrix} = \begin{bmatrix} 1 & 0 \ 0 & 1 \end{bmatrix} \] The result is the identity matrix, confirming that the calculated inverse is correct.

Key Concepts

Determinant of a Matrix

The determinant of a matrix is a special number that helps in understanding some properties of the matrix, such as if it is invertible. For a 2x2 matrix, the determinant can be easily calculated. It serves as a quick check to see whether we can find the inverse of a matrix or not.
In general, the determinant of a 2x2 matrix \( \begin{bmatrix} a & b \ c & d \end{bmatrix} \) is given by the formula:

• \( \text{det}(A) = ad - bc \)

If this determinant is not zero, it indicates that the matrix is invertible.
In our example, the matrix \( \begin{bmatrix} 1 & 3 \ 2 & 5 \end{bmatrix} \) has a determinant of \(-1\), meaning it is invertible. The negative and non-zero value here does not matter for invertibility, as any non-zero value indicates the matrix can be inverted.

2x2 Matrix

A 2x2 matrix is one that has two rows and two columns. This type of matrix is particularly nice to work with for a few reasons:

• It is simple and easy to handle because it has fewer numbers to work with compared to larger matrices.
• Both the determinant and the inverse have straightforward methods for computation.

The matrix \( \begin{bmatrix} 1 & 3 \ 2 & 5 \end{bmatrix} \) is an example of a 2x2 matrix. Its simplicity allows you to quickly perform operations like finding its determinant or inverse using standard formulas.
Dealing with a 2x2 matrix is essentially where many students start learning about the concept of matrices, making it a critical stepping stone in understanding more complex structures.

Invertible Matrix

An invertible matrix, sometimes called a non-singular matrix, is one that has an inverse. Not all matrices are invertible, though. In general, for a matrix to be invertible, its determinant must not be zero.
For the 2x2 matrix \( A = \begin{bmatrix} 1 & 3 \ 2 & 5 \end{bmatrix} \), its determinant is \(-1\). Since this value is non-zero, we confirm that \( A \) is invertible.
The inverse of a 2x2 matrix \( \begin{bmatrix} a & b \ c & d \end{bmatrix} \) can be calculated with the formula:

• \( A^{-1} = \frac{1}{\text{det}(A)} \begin{bmatrix} d & -b \ -c & a \end{bmatrix} \)

Applying this to our matrix \( A \), we find that\( A^{-1} = \begin{bmatrix} -5 & 3 \ 2 & -1 \end{bmatrix} \). Having an inverse is a valuable property because it helps in solving equations that involve matrices.

Identity Matrix

The identity matrix acts as the number one in matrix mathematics. When you multiply any matrix by an identity matrix, the initial matrix remains unchanged.
For a 2x2 matrix, the identity matrix is:

• \( I = \begin{bmatrix} 1 & 0 \ 0 & 1 \end{bmatrix} \)

Multiplying a matrix by its inverse should result in the identity matrix. This property is what you use to check if you have calculated the inverse correctly.
In our example, when we multiplied \( A \) by its inverse, \( A^{-1} \), we got:

• \( A \cdot A^{-1} = \begin{bmatrix} 1 & 3 \ 2 & 5 \end{bmatrix} \cdot \begin{bmatrix} -5 & 3 \ 2 & -1 \end{bmatrix} = \begin{bmatrix} 1 & 0 \ 0 & 1 \end{bmatrix} \)

This result verifies that the calculations were correct, as the product is indeed the identity matrix.