Problem 12 - Algebra and Trigonometry

Question

Find the inverse of the matrix if it exists. $$ \left[\begin{array}{rr}{-7} & {4} \\ {8} & {-5}\end{array}\right] $$

Step-by-Step Solution

Answer

The inverse is \( \begin{bmatrix} -\frac{5}{3} & -\frac{4}{3} \\ -\frac{8}{3} & -\frac{7}{3} \end{bmatrix} \).

Step 1: Verify the Matrix is Square

Check that the given matrix \( A = \begin{bmatrix} -7 & 4 \ 8 & -5 \end{bmatrix} \) is a square matrix. Since it is 2x2, it qualifies for finding an inverse.

Step 2: Calculate the Determinant

For a 2x2 matrix \( A = \begin{bmatrix} a & b \ c & d \end{bmatrix} \), the determinant \( \det(A) \) is calculated as \( ad - bc \). Substituting the values:\[det(A) = (-7)(-5) - (4)(8) = 35 - 32 = 3\]Since the determinant is non-zero (3), the inverse exists.

Step 3: Determine the Inverse Formula

The inverse of a 2x2 matrix \( A = \begin{bmatrix} a & b \ c & d \end{bmatrix} \) is given by:\[A^{-1} = \frac{1}{ad-bc} \begin{bmatrix} d & -b \ -c & a \end{bmatrix}\]Substitute \( a = -7 \), \( b = 4 \), \( c = 8 \), and \( d = -5 \) into the formula.

Step 4: Calculate the Inverse

Applying the values into the inverse formula:\[A^{-1} = \frac{1}{3} \begin{bmatrix} -5 & -4 \ -8 & -7 \end{bmatrix} = \begin{bmatrix} -\frac{5}{3} & -\frac{4}{3} \ -\frac{8}{3} & -\frac{7}{3} \end{bmatrix}\]This is the inverse of the original matrix.

Key Concepts

Determinant

The determinant is a special number that you can calculate from a square matrix. For a 2x2 matrix, such as the one in the exercise, the determinant helps us determine if the inverse of that matrix exists. Let's consider a generic 2x2 matrix:\[A = \begin{bmatrix} a & b \ c & d \end{bmatrix}\]The formula for the determinant of this matrix is:- \(\det(A) = ad - bc\)Here, \( a, b, c, \) and \( d \) are the numbers that fill out the matrix. By calculating the determinant, we can find out whether the matrix is invertible. If the determinant equals zero, the matrix does not have an inverse. In our case, the matrix had a determinant of 3, which is not zero, so we can proceed with finding the inverse.

2x2 Matrix

In linear algebra, matrices are arrays of numbers. A 2x2 matrix is a basic form where there are two rows and two columns. This small matrix can perform various calculations. Matrices like this are square, meaning they have the same number of rows and columns. Just as with numbers, matrices have operations such as addition, subtraction, and multiplication. The inverse operation is specifically linked to matrices. Finding the inverse of a matrix, however, requires it to be square and have a non-zero determinant.For a 2x2 matrix:- \(\begin{bmatrix} a & b \ c & d \end{bmatrix}\)If its determinant is non-zero, its inverse can be found. The inverse is useful in solving linear equations where the matrix is a coefficient. It acts similarly to finding the reciprocal of a number.

Linear Algebra

Linear Algebra is a branch of mathematics dealing with vectors, matrices, and linear equations. It's a fundamental area that lays the groundwork for advanced studies in mathematics and engineering. Many concepts in Linear Algebra revolve around matrices and their properties, like determinants and inverses. Here's why Linear Algebra is crucial:

• It allows solving systems of linear equations efficiently.
• It finds applications in areas like physics, computer science, economics, and statistics.
• Matrix inverses in Linear Algebra help in reversing operations, similar to division in arithmetic.

In our exercise, the matrix inverse was calculated—a common task in Linear Algebra. This operation is indispensable in fields like computer graphics and optimization, where pivoting transformations and changing perspectives are routine.