Problem 8 - Algebra and Trigonometry Real Mathematics, Real People

Question

The Inverse of a Matrix, show that \(B\) is the inverse of \(A\). $$A=\left[\begin{array}{rr} -\frac{1}{2} & -\frac{5}{4} \\ 1 & 2 \end{array}\right], \quad B=\left[\begin{array}{rr} 8 & 5 \\ -4 & -2 \end{array}\right]$$

Step-by-Step Solution

Answer

Arrowing from the steps, it can be concluded that indeed, matrix B is the inverse of matrix A.

Step 1: Matrix multiplication A with B (AB)

The multiplication of two matrices is, in this case A=[-1/2 & -5/4 \ 1 & 2] and B=[8 & 5 \ -4 & -2], obtained by computing the sums of the products of the corresponding entries. The entry in the i-th row and j-th column of AB is the sum of the product of entries in the i-th row of A with the corresponding entry in the j-th column of B. AB=[(-1/2*8 + -5/4*-4) & (-1/2*5 + -5/4*-2) \ (1*8 + 2*-4) & (1*5 + 2*-2)] = [1 & 0 \ 0 & 1]

Step 2: Matrix multiplication B with A (BA)

Similarly, for BA, it is computed by BA=[(8*-1/2 + 5*1) & (8*-5/4 + 5*2) \ (-4*-1/2 + -2*1) & (-4*-5/4 + -2*2)] = [1 & 0 \ 0 & 1]

Step 3: Check if AB=BA=Identity Matrix

From the multiplications AB and BA, both results were the Identity Matrix of size 2x2. Thus, B is the inverse of A.

Key Concepts

Matrix multiplication

Matrix multiplication is a fundamental operation in linear algebra, where two matrices combine to produce a new matrix. The process is somewhat like how numbers multiply but involves arranging multiple sums of products. For two matrices, say matrix \( A \) and matrix \( B \), you calculate the product \( AB \) by taking the rows of the first matrix and the columns of the second matrix. Each element in the resulting matrix is found by

• multiplying corresponding elements of a row from \( A \) with a column from \( B \)
• then summing those products.

For example, if matrix \( A \) is 2x2 and matrix \( B \) is also 2x2, their product will also be a 2x2 matrix. Specifically, to find the element in the first row and first column of the product \( AB \), multiply the elements of the first row of \( A \) and first column of \( B \) and sum them. Suppose:\[ A = \begin{bmatrix} -\frac{1}{2} & -\frac{5}{4} \ 1 & 2 \end{bmatrix}, B = \begin{bmatrix} 8 & 5 \ -4 & -2 \end{bmatrix} \] Then, the product matrix \( AB \) is achieved by calculating:

• First element (first row, first column): \(-\frac{1}{2} \times 8 + (-\frac{5}{4} \times -4) = 1\)
• Second element: \(-\frac{1}{2} \times 5 + (-\frac{5}{4} \times -2) = 0\)
• Third element: \(1 \times 8 + 2 \times -4 = 0\)
• Fourth element: \(1 \times 5 + 2 \times -2 = 1\)

This gives the identity matrix \( \begin{bmatrix} 1 & 0 \ 0 & 1 \end{bmatrix} \), confirming our matrix operations.

Identity matrix

The identity matrix is an incredibly important concept in matrix algebra. Think of it as the number 1 for matrices since it plays the same role in matrix multiplication that the number 1 does for numbers. If you multiply any matrix \( C \) by an identity matrix \( I \), the matrix \( C \) remains unchanged. An identity matrix is a square matrix with ones on the diagonal and zeros elsewhere. In notation, if \( I \) is an \( n \times n \) identity matrix, it looks like this:\[I = \begin{bmatrix}1 & 0 & \cdots & 0 \0 & 1 & \cdots & 0 \\vdots & \vdots & \ddots & \vdots \0 & 0 & \cdots & 1\end{bmatrix}\] In our example, when matrices \( A \) and \( B \) are multiplied in either order (\( AB \) or \( BA \)), the result is \[ \begin{bmatrix} 1 & 0 \ 0 & 1 \end{bmatrix} \], which is a 2x2 identity matrix. This tells us that B is indeed the inverse of A.

Matrix algebra

Matrix algebra is a broader field that encompasses several operations involving matrices, much like regular algebra involves numbers. The aim is to solve systems of linear equations efficiently and to manipulate linear transformations conveniently. One of the chief operations in matrix algebra is finding inverses of matrices. For any square matrix \( A \), an inverse matrix \( B \) such that \( AB = I \) and \( BA = I \) exists if \( A \) is invertible. Essentially, \( B \) "undoes" the transformation enacted by \( A \), similar to how dividing by a number serves to undo its multiplication. Understanding these concepts helps in grasping more complex topics in linear algebra.

• Matrix multiplication allows combinations of transformations.
• The identity matrix represents a "do nothing" operation.
• Matrix inverses enable solving equations for unknown transformations.

Together, these components form the core of linear algebra, an essential tool in mathematics, engineering, computer science, and physics. It simplifies complex transformations into manageable equations.