Problem 128 - Algebra and Trigonometry Real Mathematics, Real People

Question

Find the inverse of the matrix. \(\left[\begin{array}{rr}-1 & -4 \\ 1 & 2\end{array}\right]\)

Step-by-Step Solution

Answer

The inverse of the matrix is \(\left[\begin{array}{rr}1 & 2 \ -0.5 & -0.5\end {array}\right]\).

Step 1: Calculate the determinant

The determinant of a 2x2 matrix \(\left[\begin{array}{cc}a & b \ c & d\end {array}\right]\) is calculated as \(ad-bc\). Therefore, for the given matrix \[ \left[\begin{array}{rr}-1 & -4 \ 1 & 2\end{array}\right] \] the determinant will be \((-1*2) - (-4*1) = -2 - (-4) = 2.\)

Step 2: Find the adjugate matrix

The adjugate of a 2x2 matrix \(\left[\begin{array}{cc}a & b \ c & d\end {array}\right]\) is \(\left[\begin{array}{rr}d & -b \ -c & a\end {array}\right]\). Therefore, the adjugate of our matrix will be \(\left[\begin{array}{rr}2 & 4 \ -1 & -1\end{array}\right]\). Here, elements a and d are swapped, and elements b and c are negated.

Step 3: Calculate the inverse

The inverse of the matrix is computed as \(A^{-1} = \frac{1}{ad-bc}\) times the adjugate of A. Substituting in our values gives: \(\frac{1}{2}\left[\begin{array}{rr}2 & 4 \ -1 & -1\end {array}\right] = \left[\begin{array}{rr}1 & 2 \ -0.5 & -0.5\end {array}\right]\).

Key Concepts

Determinant of a Matrix

Understanding the determinant of a matrix is essential when delving into linear algebra, particularly when it comes to finding the inverse of a matrix. The determinant is a scalar value that is a function of the entries of a square matrix. It can be viewed as a measure of a matrix's 'scaling factor' for the area or volume transformation it represents. For a 2x2 matrix, such as the one given in the exercise \( \left[\begin{array}{cc}a & b \ c & d\end {array}\right] \), the determinant is computed using a simple formula: \(ad-bc\).

In simple terms, the determinant helps you figure out if the matrix has an inverse, is a singular matrix (which means it does not have an inverse), or when multiplied by another matrix, whether it preserves the area or volume. If the determinant is zero, the matrix cannot be inverted, which is a crucial concept for solving systems of linear equations. This step is the foundation for determining whether we can proceed to find the matrix's inverse.

Adjugate Matrix

The adjugate matrix, sometimes referred to as the 'adjoint' matrix, is closely related to the inverse of a matrix. To put it simply, the adjugate is the transpose of the cofactor matrix. In the case of our 2x2 matrix exercise, finding the adjugate involves a couple of swaps and sign changes. Specifically, for a matrix \( \left[\begin{array}{cc}a & b \ c & d\end {array}\right] \), the adjugate \( \left[\begin{array}{cc}d & -b \ -c & a\end {array}\right] \) is obtained by interchanging the positions of \(a\) and \(d\), and changing the signs of \(b\) and \(c\).

This operation prepares the matrix for the final step in finding its inverse. The adjugate matrix plays a central role in this calculation, as it is used to adjust each term when the matrix is multiplied by the reciprocal of its determinant. This adjustment ensures that when this new matrix is multiplied with the original, the result will be the identity matrix, confirming the properties of inverse matrices.

Algebraic Operations on Matrices

Algebraic operations on matrices follow specific rules that are different from those applied to regular numbers. Key operations include addition, subtraction, multiplication, and finding the inverse. When performing these operations, especially multiplication and inversion, it's important to adhere to the defined protocols.For matrix multiplication, the number of columns in the first matrix must equal the number of rows in the second matrix for the operation to be possible. And, when calculating the inverse, as demonstrated in the previous steps, we first determine if the matrix is invertible by checking if its determinant is non-zero.

Once the determinant is found and deemed non-zero, we utilize the adjugate matrix and multiply each of its elements by \( 1 / determinant\), as explained in the given exercise. It's these combinations of algebraic operations that allow us to find the inverse of a matrix, which has a plethora of applications, including in solving systems of linear equations, in computer graphics for transformations, and in encryption algorithms.