Problem 61 - A Graphical Approach to College Algebra

Question

Given a square matrix \(A^{-1}\), find matrix \(A\). $$A^{-1}=\left[\begin{array}{rr} 5 & -9 \\ -1 & 2 \end{array}\right]$$

Step-by-Step Solution

Answer

The original matrix \(A\) is \(\begin{bmatrix} 2 & 9 \\ 1 & 5 \end{bmatrix}\).

Step 1: Understanding Inverse Matrix Property

To find the original matrix \(A\) from its inverse \(A^{-1}\), we use the property that \(A \cdot A^{-1} = I\), where \(I\) is the identity matrix. For a 2x2 matrix, the identity matrix is \(\begin{bmatrix} 1 & 0 \ 0 & 1 \end{bmatrix}\). Therefore, we need to find \(A\) such that \(A \cdot A^{-1} = \begin{bmatrix} 1 & 0 \ 0 & 1 \end{bmatrix}\).

Step 2: Determinant Calculation

Given the inverse matrix \(A^{-1} = \begin{bmatrix} 5 & -9 \ -1 & 2 \end{bmatrix}\), we need to first find the determinant \(det(A)\) of the original matrix \(A\) because \(A^{-1} = \frac{1}{det(A)} adj(A)\), where \(adj(A)\) is the adjugate of \(A\). From here, we can deduce certain properties about \(det(A)\) by reversing the formula for 2x2 matrices.

Step 3: Adjugate of A

The adjugate of matrix \(A\) is computed as \(adj(A) = \begin{bmatrix} d & -b \ -c & a \end{bmatrix}\) for a 2x2 matrix \(A = \begin{bmatrix} a & b \ c & d \end{bmatrix}\). In this case, since \(A^{-1} = \begin{bmatrix} 5 & -9 \ -1 & 2 \end{bmatrix}\), this is actually the adjugate matrix, divided by \(det(A)\). Therefore, setting \(adj(A) = A^{-1}\), gives us \(adj(A) = \begin{bmatrix} 1 & 0 \ 0 & 1 \end{bmatrix} \times det(A)\).

Step 4: Calculate Original Matrix A

Since \(adj(A) = A^{-1} \times det(A)\), we solve for \(A\) using the inverse formula. We find \(A = \frac{1}{5 \times 2 - (-9) \times (-1)} \begin{bmatrix} 2 & 9 \ 1 & 5 \end{bmatrix}\). Simplify the determinant \((2 \cdot 5 - (-9 \cdot -1)) = 10 - 9 = 1\). So \(A = \begin{bmatrix} 2 & 9 \ 1 & 5 \end{bmatrix}\).

Key Concepts

Inverse Matrix

The inverse of a matrix is often denoted as \(A^{-1}\). If you multiply a matrix by its inverse, the result will be the identity matrix. This property comes in handy, especially in solving equations or reverting linear transformations.

Here is the catch: **not all matrices have inverses**! A matrix can only have an inverse if it satisfies certain conditions, specifically, having a non-zero determinant.

To find the inverse of a 2x2 matrix \(\begin{bmatrix} a & b \ c & d \end{bmatrix}\), the inverse, if it exists, is given by: \[ A^{-1} = \frac{1}{ad-bc}\begin{bmatrix} d & -b \ -c & a \end{bmatrix} \] The expression \(ad - bc\) is known as the determinant, which we'll cover next.

Determinant

The determinant of a matrix is a special number, which is particularly crucial for understanding the properties of a matrix. Think of the determinant as a value that determines if a matrix is invertible.

For a 2x2 matrix \(\begin{bmatrix} a & b \ c & d \end{bmatrix}\), the determinant is calculated as \(ad - bc\).

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

Why? Because the determinant, quite literally, measures how "stretched" or "squished" the area covered by a matrix transformation is. No stretch (or zero value) means it's not covering any new ground, hinting at non-invertibility.

Identity Matrix

The identity matrix acts as the building block of matrices, much like the number 1 in multiplication. Its key role is that any matrix multiplied by the identity matrix remains unchanged. This property is crucial when discussing inverses.

• The identity matrix is typically denoted as \(I\).
• For any matrix \(A\), \(A \cdot I = I \cdot A = A\).

In 2x2 matrices, the identity matrix looks like \(\begin{bmatrix} 1 & 0 \ 0 & 1 \end{bmatrix}\). You can see how it "does nothing" to the matrix it multiplies, maintaining the original values intact.

When finding an inverse, the ultimate goal is to check if the product of the matrix and its inverse results in this special identity matrix.

Adjugate

The adjugate is an essential component when calculating the inverse of a matrix. It is essentially the transpose of the cofactor matrix. Its usefulness becomes evident when you don't have the matrix inverse directly at hand.

For a 2x2 matrix \(\begin{bmatrix} a & b \ c & d \end{bmatrix}\), the adjugate is constructed as follows:
\[ adj(A) = \begin{bmatrix} d & -b \ -c & a \end{bmatrix} \]
The adjugate serves as part of the formula to find the inverse matrix. Since the inverse \(A^{-1}\) is calculated as \(\frac{1}{det(A)} adj(A)\), the adjugate outlines all the necessary adjustments needed to flip the roles in the matrix.