Problem 71 - College Algebra and Calculus: An Applied Approach

Question

Find a matrix \(B\) such that \(A B\) is the identity matrix. Is there more than one correct result? $$ A=\left[\begin{array}{ll} 1 & 3 \\ 1 & 2 \end{array}\right] $$

Step-by-Step Solution

Answer

The matrix \(B\) that gives the identity matrix when multiplied with \(A\) is \[\left[\begin{array}{cc}-2 & 3\ 1 & -1\end{array}\right]\]. There is only one correct result for this problem, as the inverse of a matrix is unique.

Step 1: Calculate the Determinant of A

First, calculate the determinant of the given matrix \(A\). The determinant of a 2x2 matrix \[\left[\begin{array}{cc}a & b\ c & d\end{array}\right]\] is given by \(ad-bc\). So, for matrix \(A\), it is \(1*2 - 3*1 = -1\).

Step 2: Find the Adjugate Matrix

Next, find the adjugate of \(A\). The adjugate is found by swapping the elements on the leading diagonal, and changing the signs of the other two elements. So, for \(A\), the adjugate matrix is \[\left[\begin{array}{cc}2 & -3\ -1 & 1\end{array}\right]\].

Step 3: Calculate the Inverse matrix B

The final step is to calculate the inverse matrix by dividing the adjugate matrix by the determinant. Thus, the inverse matrix \(B\) is \[\left[\begin{array}{cc}\frac{-2}{1} & \frac{3}{1}\ \frac{1}{1} & \frac{-1}{1}\end{array}\right]\], which simplifies to \[\left[\begin{array}{cc}-2 & 3\ 1 & -1\end{array}\right]\].

Key Concepts

Determinant

The determinant is a special number that you can calculate from a square matrix. When dealing with a 2x2 matrix, the calculation is straightforward. Suppose you have a matrix \( \begin{bmatrix} a & b \ c & d \end{bmatrix} \). To find its determinant, you use the formula \( ad - bc \). In simple terms, multiply the top-left element by the bottom-right element and subtract the product of the top-right and bottom-left elements.
This value is essential when working with matrices, especially when finding an inverse matrix. If the determinant is zero, it means the matrix doesn't have an inverse. In our exercise, we calculated the determinant of matrix \( A \) as \(-1\), which is non-zero. This means \( A \) has an inverse matrix, allowing us to find matrix \( B \) such that \( A \times B \) equals the identity matrix.

Adjugate

The adjugate of a matrix is the transpose of its cofactor matrix. For a 2x2 matrix, it's easier to remember the method:

• Swap the elements on the diagonal (top-left and bottom-right).
• Change the signs of the off-diagonal elements (top-right and bottom-left).

In our example, we took matrix \( A \) which is \( \begin{bmatrix} 1 & 3 \ 1 & 2 \end{bmatrix} \) and found its adjugate by swapping 1 and 2, and changing the signs of 3 and 1. This resulted in the adjugate matrix \( \begin{bmatrix} 2 & -3 \ -1 & 1 \end{bmatrix} \).
Using the adjugate matrix is important because it helps us construct the inverse of a matrix by dividing each element of the adjugate by the determinant.

Identity Matrix

The identity matrix is a type of matrix that acts as the 'one' in matrix arithmetic. It doesn't change a matrix when multiplied by it, similar to how multiplying any number by 1 does not change its value.
For a 2x2 matrix, the identity matrix is \( \begin{bmatrix} 1 & 0 \ 0 & 1 \end{bmatrix} \). This matrix is special because multiplying any 2x2 matrix by the identity matrix will leave the original matrix unchanged. In our task, we aim to find a matrix \( B \) such that \( A \times B = I \), where \( I \) is the identity matrix.

• It assures us of the concept of reversibility in matrix transformations.
• Finding such a matrix \( B \) ensures that \( A \) has an inverse and is invertible.

2x2 Matrix

A 2x2 matrix is a small, square array that consists of two rows and two columns. It looks like this: \( \begin{bmatrix} a & b \ c & d \end{bmatrix} \). Despite its simplicity, it holds a lot of utility in mathematics, especially in linear transformations and systems of equations.
Working with a 2x2 matrix is often a starting point when learning about more complicated matrices. Its size allows for elegant formulas, such as for finding the determinant and inverse, which become more complex for larger matrices.

• 2x2 matrices are particularly important in describing linear transformations in 2D space.
• They form the basis of understanding higher-dimension matrices and their properties.

In our exercise, the matrix \( A \) was a 2x2 matrix, and finding its inverse involved using straightforward operations including calculating the determinant and then the adjugate.