Problem 71 - Algebra 2

Question

Find the inverse of each matrix, if it exists. $$ \left[\begin{array}{rr}{2} & {5} \\ {-1} & {-2}\end{array}\right] $$

Step-by-Step Solution

Answer

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

Step 1: Understand the Concept of Inversion

To find the inverse of a matrix \( A \), the matrix must be square and have a non-zero determinant. For a 2x2 matrix \( A = \begin{bmatrix} a & b \ c & d \end{bmatrix} \), the inverse is given by \( A^{-1} = \frac{1}{ad-bc} \begin{bmatrix} d & -b \ -c & a \end{bmatrix} \), provided the determinant \( ad-bc eq 0 \).

Step 2: Calculate the Determinant

Given matrix \( A = \begin{bmatrix} 2 & 5 \ -1 & -2 \end{bmatrix} \), calculate the determinant using the formula \( ext{det}(A) = (2)(-2) - (5)(-1) \). Compute the determinant: \(-4 + 5 = 1\).

Step 3: Verify Non-zero Determinant

The determinant found in Step 2 is 1, which is non-zero. Hence, the inverse exists.

Step 4: Apply the Inverse Formula

Using the formula for the inverse of a 2x2 matrix, substitute into \( A^{-1} = \frac{1}{1} \begin{bmatrix} -2 & -5 \ 1 & 2 \end{bmatrix} \). Simplify to \( A^{-1} = \begin{bmatrix} -2 & -5 \ 1 & 2 \end{bmatrix} \).

Step 5: Confirm the Result

Multiply the computed inverse \( A^{-1} = \begin{bmatrix} -2 & -5 \ 1 & 2 \end{bmatrix} \) by the original matrix \( A = \begin{bmatrix} 2 & 5 \ -1 & -2 \end{bmatrix} \) to ensure the product is the identity matrix \( I = \begin{bmatrix} 1 & 0 \ 0 & 1 \end{bmatrix} \). Performing the matrix multiplication confirms the result as the identity matrix, verifying the calculated inverse is correct.

Key Concepts

Matrix Determinant

The matrix determinant is a special number associated with square matrices. It provides important information about the matrix, especially when finding a matrix inverse.
For a 2x2 matrix, the determinant can be calculated using the formula:

• Given matrix \( A = \begin{bmatrix} a & b \ c & d \end{bmatrix} \), the determinant \( \text{det}(A) \) is \( ad - bc \).

A non-zero determinant is crucial for a matrix to have an inverse.
In our example matrix,

• \( A = \begin{bmatrix} 2 & 5 \ -1 & -2 \end{bmatrix} \),
• the determinant is \( 2(-2) - (5)(-1) = -4 + 5 = 1 \).

Since this value is non-zero, the matrix is invertible.

2x2 Matrix

A 2x2 matrix is a simple square matrix with two rows and two columns. It's the most basic type of square matrix, which makes it perfect for learners starting with matrix algebra. These matrices are often denoted as:

• \( \begin{bmatrix} a & b \ c & d \end{bmatrix} \)

The straightforward structure of 2x2 matrices makes calculations, like determining determinants and inverses, more manageable than larger matrices.
For example, with matrix \( A = \begin{bmatrix} 2 & 5 \ -1 & -2 \end{bmatrix} \), we can easily apply formulas thanks to its simple structure. Working with 2x2 matrices is an ideal way to grasp fundamental matrix concepts.

Identity Matrix

The identity matrix plays a crucial role in matrix operations. It's like the number 1 for matrices, meaning any matrix multiplied by it remains unchanged.
In the 2x2 case, the identity matrix \( I \) is represented as:

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

When confirming that a computed inverse is correct, you multiply the original matrix by its inverse. If the product is the identity matrix, the inverse is correct. In our exercise, after finding the inverse of matrix \( A \), multiplying \( A \) by \( A^{-1} \) results in the identity matrix, confirming the solution.

Invertible Matrix

An invertible matrix, also known as a non-singular matrix, is a matrix that has an inverse. This is only possible if the matrix's determinant is non-zero.
Here's what you need to know about invertible matrices:

• A square matrix must have a non-zero determinant to be invertible. Otherwise, it’s called singular and has no inverse.
• For a 2x2 matrix, the condition for invertibility is the determinant \( ad - bc eq 0 \).

In our task, since matrix \( A = \begin{bmatrix} 2 & 5 \ -1 & -2 \end{bmatrix} \) has a determinant of 1, it is non-singular, and thus invertible. This confirms that we can apply the inverse formula to find its inverse.