Problem 43 - Algebra 2

Question

The \(\underline{x^{-1}}\) key on a TI-83/84 Plus graphing calculator is used to find the inverse of a matrix. If you get a SINGULAR MATRIX error on the screen, then the matrix has no inverse. Find the inverse of each matrix. $$ \left[\begin{array}{cc}{12} & {4} \\ {15} & {5}\end{array}\right] $$

Step-by-Step Solution

Answer

The matrix is singular and has no inverse.

Step 1: Calculate the Determinant

The first step in finding the inverse of a matrix is to calculate its determinant. For a 2x2 matrix \(\begin{bmatrix} a & b \ c & d \end{bmatrix}\), the determinant \( \text{det}(A) \) is given by \(ad - bc\). In this case, the matrix is \(\begin{bmatrix} 12 & 4 \ 15 & 5 \end{bmatrix}\). So, the determinant is calculated as \(12 \times 5 - 15 \times 4 = 60 - 60 = 0\).

Step 2: Check Determinant

If the determinant of the matrix is zero, then the matrix is singular and does not have an inverse. In this particular case, we found that the determinant is zero.

Key Concepts

Determinant

In the matrix world, the determinant is a special number. It helps us understand certain properties of matrices. For a 2x2 matrix, which is quite common, its determinant is calculated as follows: If you have a matrix \( \begin{bmatrix} a & b \ c & d \end{bmatrix} \), the determinant, denoted as \( \text{det}(A) \), is calculated by \(ad - bc\).

The determinant is crucial for finding if a matrix is invertible. The inverse of a matrix is like the matrix's opposite. When you multiply a matrix by its inverse, you should end up with the identity matrix, just like multiplying a number by its reciprocal equals one.

If the determinant is zero, however, the matrix is called singular. Singular matrices are special because they don’t have an inverse. This is similar to how the number zero doesn’t have a reciprocal in regular arithmetic.

Singular Matrix

A singular matrix is a matrix that does not have an inverse. This happens if the determinant of the matrix is zero. When you try to find the inverse of a matrix and see "SINGULAR MATRIX" on a graphing calculator, it means something important: the calculation is stalled because the matrix cannot be reversed.

For example, the matrix \(\begin{bmatrix} 12 & 4 \ 15 & 5 \end{bmatrix}\) is singular. We found this out because its determinant was zero. Matrices become singular for various reasons:

• Rows or columns are proportional, like duplicates.
• All elements add up to zero in a row or column.
• Matrix operations result in zero rows or columns.

Singular matrices are problematic in computations where an inverse is necessary. Understanding this key concept helps when handling larger matrix problems and ensuring the correct methods are applied.

Graphing Calculator

Graphing calculators like the TI-83/84 Plus are powerful tools. They assist in computations involving matrices, among many other functions. The \( \underline{x^{-1}} \) button on these calculators is specifically designed to help find the inverse of a matrix quickly.

Here's how you use it:

• Input your matrix using the matrix function key.
• Select the matrix you've just inputted.
• Press the \( \underline{x^{-1}} \) key to calculate the inverse.

Keep in mind, if the matrix is singular, the calculator will display a "SINGULAR MATRIX" error. This error indicates that you can't calculate the inverse because the matrix's determinant is zero.

Using graphing calculators can greatly simplify the process of dealing with matrices. They provide instant feedback about matrix properties, allowing you to understand the characteristics of matrices without manually doing all the calculations.