Problem 7 - Algebra and Trigonometry Real Mathematics, Real People

Question

The Inverse of a Matrix, show that \(B\) is the inverse of \(A\). $$A=\left[\begin{array}{ll} 5 & 4 \\ 3 & 2 \end{array}\right], \quad B=\left[\begin{array}{rr} -1 & 2 \\ \frac{3}{2} & -\frac{5}{2} \end{array}\right]$$

Step-by-Step Solution

Answer

Yes, B is the inverse of A as multiplication of A and B in either order results in an identity matrix.

Step 1: Multiply A and B

A*B = \( \left[\begin{array}{ll} 5 & 4 \ 3 & 2 \end{array}\right] * \left[\begin{array}{rr}-1 & 2 \rac{3}{2} & -\frac{5}{2} \end{array}\right] = \left[\begin{array}{cc} 5*(-1) + 4*(\frac{3}{2}) & 5*2 + 4*(-\frac{5}{2}) \ 3*(-1) + 2*(\frac{3}{2}) & 3*2 + 2*(-\frac{5}{2}) \end{array}\right] \)

Step 2: Perform the calculations

The matrix after multiplying A and B= \(\left[\begin{array}{cc} -5 + 6 & 10 - 10 \ -3 + 3 & 6 - 5 \end{array}\right] = \left[\begin{array}{cc} 1 & 0 \ 0 & 1 \end{array}\right]\)

Step 3: Multiply B and A

B*A = \( \left[\begin{array}{rr}-1 & 2 \rac{3}{2} & -\frac{5}{2} \end{array}\right] * \left[\begin{array}{ll} 5 & 4 \ 3 & 2 \end{array}\right] = \left[\begin{array}{cc} -1*5 + 2*3 & -1*4 + 2*2 \ \frac{3}{2}*5 + (-\frac{5}{2})*3 & \frac{3}{2}*4 + (-\frac{5}{2})*2 \end{array}\right] \)

Step 4: Perform the calculations

The matrix after multiplying B and A= \(\left[\begin{array}{cc} -5 + 6 & -4 + 4 \ \frac{15}{2} - \frac{15}{2} & 6 - 5 \end{array}\right] = \left[\begin{array}{cc} 1 & 0 \ 0 & 1 \end{array}\right]\)

Step 5: Assess if B is the Inverse of A

As can be seen from steps 2 and 4, multiplying matrices A and B, in either order A*B or B*A, results in an identity matrix. Hence, it can be confirmed that matrix B is definitely the inverse of matrix A.

Key Concepts

Matrix Multiplication

Matrix multiplication is a fundamental operation in linear algebra, allowing us to transform one matrix by another. To multiply two matrices, we follow a specific procedure. Suppose we have two matrices, such as \( A \) and \( B \). Each element of the resulting matrix is calculated by multiplying the rows of the first matrix by the columns of the second matrix.

Think of matrix multiplication as a series of dot products. For each entry, multiply corresponding elements from the row of the first matrix with the column of the second, then sum them up. This procedure is repeated for each row and column.

It's important to remember the condition for multiplication: the number of columns in the first matrix must match the number of rows in the second matrix. Otherwise, the matrices cannot be multiplied.

• Order of multiplication matters. Generally, \( A \times B eq B \times A \).
• The size of the resulting matrix is determined by the rows of the first and the columns of the second matrix.

Understanding matrix multiplication is key when working with matrix operations, including finding the inverse.

Identity Matrix

The identity matrix is like the number 1 in matrix math. Multiplying any matrix by the identity matrix leaves it unchanged, just like multiplying a number by 1. It is a square matrix where all the diagonal elements are 1, and all other elements are 0.

For example, a 2x2 identity matrix looks like this:\[I = \begin{bmatrix} 1 & 0 \ 0 & 1 \end{bmatrix}\]

When you multiply any matrix, say \( A \), by an identity matrix, it satisfies the equation \( A \times I = A \). Similarly, multiplying in reverse order, \( I \times A = A \).

The identity matrix is central in confirming the inverse of a matrix. When a matrix \( A \) is multiplied by its inverse \( B \), the result is the identity matrix:

• \( A \times B = I \)
• \( B \times A = I \)

This property helps us verify the correctness of matrix computations, especially finding inverses.

Inverse of a Matrix

The inverse of a matrix, similar to the reciprocal of a number, "undoes" the effect of the original matrix. If \( B \) is the inverse of \( A \), multiplying \( A \) by \( B \) yields the identity matrix.

For a matrix \( A \), its inverse \( A^{-1} \) is defined such that:

• \( A \times A^{-1} = I \)
• \( A^{-1} \times A = I \)

Finding the inverse involves specific conditions: not all matrices have inverses. A matrix must be square (same number of rows and columns) and have a non-zero determinant. The process to find an inverse is methodical but checks these conditions first.

In the given exercise, multiplying matrix \( A \) by matrix \( B \), and vice versa, results in the identity matrix. This verifies that \( B \) is indeed the inverse of \( A \), showcasing the importance of matrix multiplication and the identity matrix in understanding inverses.