Problem 13 - 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 \(A^{-1} A=I_{2}\) $$A=\left[\begin{array}{rr}2 & 3 \\\\-1 & 2\end{array}\right]$$

Step-by-Step Solution

Answer

The inverse of matrix A is \( A^{-1} = \left[\begin{array}{cc}\frac{2}{7} & -\frac{3}{7} \ \frac{1}{7} & \frac{2}{7}\end{array}\right] \). This is checked and validated by the fact that both multiplications \( AA^{-1} \) and \( A^{-1}A \) result in the identity matrix \( I_{2} \).

Step 1: Use the formula to find the inverse

First, plug the values of a, b, c, and d into the formula for the inverse of a 2x2 matrix. In this case, a = 2, b = 3, c = -1, d = 2. The determinant is given by \( \frac{1}{ad - bc} = \frac{1}{(2)(2) - (3)(-1)} = \frac{1}{4 + 3} = \frac{1}{7} \)Use this to compute the inverse:\( A^{-1} = \frac{1}{7} \left[\begin{array}{cc}2 & -3 \ 1 & 2\end{array}\right] = \left[\begin{array}{cc}\frac{2}{7} & -\frac{3}{7} \ \frac{1}{7} & \frac{2}{7}\end{array}\right] \)

Step 2: Check \(A A^{-1}=I_{2}\)

Next, multiply matrix A by matrix \( A^{-1} \): \( A A^{-1} = \left[\begin{array}{rr}2 & 3 \-1 & 2\end{array}\right] \left[\begin{array}{cc}\frac{2}{7} & -\frac{3}{7} \ \frac{1}{7} & \frac{2}{7}\end{array}\right] = \left[\begin{array}{cc}1 & 0 \ 0 & 1\end{array}\right] \) This is the 2x2 identity matrix \( I_{2} \)

Step 3: Check \(A^{-1} A=I_{2}\)

Finally, multiply matrix \( A^{-1} \) by matrix A: \( A^{-1} A = \left[\begin{array}{cc}\frac{2}{7} & -\frac{3}{7} \ \frac{1}{7} & \frac{2}{7}\end{array}\right] \left[\begin{array}{rr}2 & 3 \-1 & 2\end{array}\right] = \left[\begin{array}{cc}1 & 0 \ 0 & 1\end{array}\right] \)Again this is the 2x2 identity matrix \( I_{2} \), which verifies the inverse matrix.

Key Concepts

2x2 matrix

A 2x2 matrix is one of the simplest forms of a matrix, which consists of two rows and two columns. Mathematically, it is represented as follows:\[A = \begin{bmatrix} a & b \ c & d \end{bmatrix}\]where \(a\), \(b\), \(c\), and \(d\) are elements of the matrix.

• Each element can be a real number or a more complex number.
• The position of each element is defined by its row and column in the matrix.

The 2x2 matrix is often used in various mathematical computations because it is simple yet powerful. It is crucial in many applications, including solving linear equations, transformations, and even graphics.

determinant

The determinant of a 2x2 matrix is a special number that provides useful properties, like checking if a matrix is invertible. For a matrix:\[A = \begin{bmatrix} a & b \ c & d \end{bmatrix}\]the determinant is calculated as \(ad - bc\).

• If the determinant is zero, the matrix cannot be inverted.
• A non-zero determinant indicates that the matrix is invertible.
• It is also crucial in finding solutions to linear systems and understanding linear transformations.

In the problem above, the determinant plays a vital role in calculating the inverse of the matrix since the inverse is only possible when the determinant is non-zero.

identity matrix

The identity matrix is a special matrix used in matrix arithmetic. For a 2x2 matrix, the identity matrix, denoted as \(I_2\), is:\[I_2 = \begin{bmatrix} 1 & 0 \ 0 & 1 \end{bmatrix}\]

• An identity matrix acts like the number 1 in regular multiplication; any matrix multiplied by the identity matrix remains unchanged.
• In computational terms, if \(A\) is a matrix, then \(AI = IA = A\).
• Verifying matrix inverses involves checking whether multiplying the matrix by its inverse results in the identity matrix.

In the given exercise, verifying that both \(AA^{-1}=I_2\) and \(A^{-1}A=I_2\) confirms that we correctly found the inverse.

matrix multiplication

Matrix multiplication is a process that combines rows of one matrix with columns of another to produce a new matrix. For two matrices, \(A\) and \(B\), the product \(AB\) can be found if the number of columns in \(A\) equals the number of rows in \(B\).

• In a 2x2 example, each element of the resulting matrix is calculated by multiplying and then summing up corresponding elements from the row of the first matrix and column of the second.
• Matrix multiplication is not commutative, meaning \(AB\) is not necessarily equal to \(BA\).
• It is extensively used in various fields including computer graphics, where transformations (such as rotation and scaling) are expressed using matrices.

In this exercise, after finding the inverse matrix, multiplying it with the original matrix led us back to the identity matrix \(I_2\), demonstrating the core property of matrix inverses.