Problem 16 - College Algebra

Question

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

Step-by-Step Solution

Answer

The inverse of matrix A is \(A^{-1}=\left[\begin{array}{rr}-1 & 3 \-0.5 & 1\end{array}\right]\). The calculations fulfill the properties of inverse matrices, confirming the correctness of the solution.

Step 1: Find the inverse matrix

Use the given formula for the inverse of a 2x2 matrix with \(A=\left[\begin{array}{ll}2 & -6 \1 & -2\end{array}\right]\). First, calculate the determinant \(ad - bc\), which here is \((2*-2) - (1*-6) = -4 + 6 = 2\). Finally, use the determinant in the formula to obtain the inverse matrix: \(A^{-1}=\frac{1}{2}\left[\begin{array}{rr}-2 & 6 \-1 & 2\end{array}\right]\) which results in \(A^{-1}=\left[\begin{array}{rr}-1 & 3 \-0.5 & 1\end{array}\right]\)

Step 2: Check the results

To confirm if the obtained matrix is indeed the inverse, multiply the original matrix A and the inverse matrix \(A^{-1}\) in both order. Firstly, compute \(A A^{-1}\) which gives \(\left[\begin{array}{ll}2 & -6 \1 & -2\end{array}\right]\left[\begin{array}{ll}-1 & 3 \-0.5 & 1\end{array}\right] = \left[\begin{array}{ll}1 & 0 \0 & 1\end{array}\right]\). Secondly, compute \(A^{-1} A\), which gives \(\left[\begin{array}{ll}-1 & 3 \-0.5 & 1\end{array}\right]\left[\begin{array}{ll}2 & -6 \1 & -2\end{array}\right] =\left[\begin{array}{ll}1 & 0 \0 & 1\end{array}\right]\). Both results are the 2x2 identity matrix, confirming that the found inverse is correct.

Key Concepts

Determinant

When exploring matrices, one key concept is the determinant. The determinant is a special number that can reveal a lot about a matrix. For a 2x2 matrix like we have here, it's calculated using the formula \( ad - bc \).
In this expression, \( a \), \( b \), \( c \), and \( d \) are the elements of the matrix arranged as follows:

• \( a \) is the top-left element,
• \( b \) is the top-right,
• \( c \) is the bottom-left,
• \( d \) is the bottom-right.

The determinant helps in finding the inverse of a matrix. If the determinant is zero, the matrix doesn't have an inverse. This is crucial because an inverse allows solving matrix equations and understanding more about transformations the matrix represents.

In our case, the determinant calculation was \((2 * -2) - (1 * -6) = -4 + 6 = 2.\) Since the determinant is non-zero, the matrix is invertible.

Identity Matrix

The identity matrix is a fundamental concept just as the number 1 is in multiplication. It is a square matrix with ones on the diagonal and zeros elsewhere. For a 2x2 matrix, it looks like this:

• \[I_{2} = \begin{bmatrix} 1 & 0 \ 0 & 1 \end{bmatrix}\]

This matrix is the multiplicative identity in matrix algebra, meaning any matrix multiplied by the identity matrix remains unchanged. This property allows us to check our work when finding the inverse of a matrix.
In our step-by-step solution, we verified our inverse matrix by confirming that both \( A \times A^{-1} \) and \( A^{-1} \times A \) resulted in the identity matrix. This not only confirms the accuracy of our inverse but also validates our calculations as consistent with matrix multiplication rules.

Matrix Multiplication

Matrix multiplication involves combining two matrices to produce a third matrix, observing specific rules concerning their dimensions. It's a bit like multiplying numbers, but we perform a series of multiplication and addition operations.
To multiply two matrices, you take each row of the first matrix and multiply it by each column of the second matrix. The result is a matrix where each element is the sum of the products of elements of the rows and columns.

• For example, to find the element in the first row, first column of the resulting matrix, multiply each element of the first row of the first matrix by the corresponding element of the first column of the second matrix and add them.
• Continue this process for each element within the result matrix.

For checking our inverse, we multiplied matrix \( A \) and its inverse \( A^{-1} \) to verify they produce the identity matrix. This ensures the operations were done correctly. Practicing matrix multiplication can help in understanding and utilizing matrices in solving algebraic problems.