Problem 18 - College Algebra

Question

In Exercises \(13-18,\) use the fact that if \(A=\left[\begin{array}{ll}a & b \\\ c & d\end{array}\right]\), then \(A^{-1}=\frac{1}{a d-b c}\left[\begin{array}{cc}d & -b \\ -c & a\end{array}\right]\) to find the inverse of cach matrix, if possible. Check that \(A A^{-1}=I_{2}\) and \(A^{-1} A=I_{2}\) $$A=\left[\begin{array}{rr}6 & -3 \\\\-2 & 1\end{array}\right]$$

Step-by-Step Solution

Answer

The given matrix A does not have an inverse because its determinant equals 0.

Step 1: Identify matrix parameters

According to given matrix, determine parameters: \(a = 6\), \(b = -3\), \(c = -2\), \(d = 1\).

Step 2: Apply the inverse formula

Apply the formula to calculate the inverse matrix. \(A^{-1} = \frac{1}{a d - b c} \left[\begin{array}{cc}d & -b \ -c & a\end{array}\right] = \frac{1}{6*1 - (-3)*(-2)} \left[\begin{array}{cc}1 & 3 \ 2 & 6\end{array}\right]\)

Step 3: Calculate the determinant and find A inverse

Calculate the determinant (\(a d - b c\)) and apply the result to find \(A^{-1}\): \(A^{-1} = \frac{1}{6 - 6} \left[\begin{array}{cc}1 & 3 \ 2 & 6\end{array}\right]\). But the determinant equals 0 so the inverse of the matrix does not exist.

Step 4: Skip next steps

Since the inverse does not exist, there is no need to check if \(AA^{-1}=I_2\) or \(A^{-1}A=I_2\).

Key Concepts

Determinant

The determinant is a special number that can be calculated from a square matrix. It is crucial when determining if a matrix is invertible or not.
The determinant of a 2x2 matrix\( \left[\begin{array}{ll} a & b \ c & d \end{array}\right] \) is calculated using the formula:

• \( \text{det}(A) = ad - bc \)

This operation provides a single numerical value.
If this value is zero, it means the matrix is singular, and it does not have an inverse.
In our example, the calculation \(6 \times 1 - (-3) \times (-2) = 6 - 6 = 0 \) shows that the determinant is zero, meaning the inverse of the matrix cannot be computed.

2x2 Matrix

A 2x2 matrix is one of the most basic forms of matrices.
It consists of two rows and two columns and can be represented in the general form:

• \( \left[\begin{array}{ll} a & b \ c & d \end{array}\right] \)

These matrices are significant as they serve as the foundational building blocks in linear algebra.
Understanding how to interact with 2x2 matrices, like finding determinants and inverses, helps in solving larger systems in mathematics and real-world applications.
2x2 matrices are used extensively in fields such as computer graphics, physics, and engineering.

Inverse Formula

The inverse of a matrix is akin to finding the reciprocal of a number.
For a 2x2 matrix \( A = \left[\begin{array}{ll} a & b \ c & d \end{array}\right] \), its inverse \( A^{-1} \) is found using:

• \( A^{-1} = \frac{1}{ad-bc} \left[\begin{array}{cc} d & -b \ -c & a \end{array}\right] \)

This equation helps "undo" the operation of the original matrix, functioning like reverse operations in multiplication.
The formula suggests a simple switch in position of elements\( a \) and \( d \), and a change in sign for elements\( b \) and \( c \), divided by the determinant \( ad-bc \).
Applying the inverse is vital in solving matrix equations, where multiplying a matrix by its inverse results in the identity matrix, \( I_2 \).

Non-Invertible Matrix

A non-invertible matrix, also known as a singular matrix, is a matrix that does not have an inverse.
The primary condition for this is having a determinant equal to zero. When \( ad-bc = 0 \), the matrix cannot be reversed because dividing by zero is undefined.For example, in the given matrix \( \left[\begin{array}{rr} 6 & -3 \ -2 & 1 \end{array}\right] \), the determinant calculation yielded zero.
This means the matrix is non-invertible.

• Consequently, operations that involve matrix inversion, such as solving linear systems using matrices, cannot be conducted with singular matrices.

Understanding non-invertible matrices is key in linear algebra because it denotes limitations in matrix operations and has implications for system solvability.