Problem 5 - Precalculus with Limits

Question

In Exercises 5-12, show that \(B\) is the inverse of \(A\). \(A = \left[ \begin{array}{r} 2 && 1 \\ 5 && 3 \end{array} \right]\), \(B = \left[ \begin{array}{r} 3 & -1 \\ -5 & 2 \end{array} \right]\).

Step-by-Step Solution

Answer

Matrix B is indeed the inverse of matrix A, because their products equal to the identity matrix.

Step 1: Understanding Matrix Inversion

The inverse of a square matrix A is another matrix B such that if they are multiplied, AB = BA = I, the identity matrix. The identity matrix is a square matrix in which all the elements of the principal (main) diagonal are ones, and all other elements are zeros

Step 2: Multiply AB and BA

Now, perform the multiplication AB and BA. The multiplication of two 2x2 matrices can be done as follows: Multiply the elements of the first row of the first matrix by the elements of the first column of the second matrix, sum them up to get the first element of the new matrix. Multiply the elements of the first row of the first matrix by the elements of the second column of the second matrix, sum them up to get the second element of the new matrix. Repeat this process with the second row of the first matrix to get the third and fourth elements of the new matrix.

Step 3: Compare Resulting Matrices with Identity Matrix

After performing the multiplications, you should compare the products to the 2x2 identity matrix, which is \[\begin{array}{r}1 & 0\0 & 1 \end{array}\]If both product matrices equal the identity matrix, this proves that B is indeed the inverse of A.

Key Concepts

Inverse Matrix

An inverse matrix is essentially the reverse of a given square matrix. Imagine it as the undoing partner of the original matrix. If matrix A has an inverse, identified as matrix B, then when these two matrices are multiplied in either order, the result will be the identity matrix.

Not all matrices have inverses. A square matrix will possess an inverse if it is non-singular, meaning it has a determinant that is not equal to zero. The formula to find the inverse of a 2x2 matrix \(A = \left[ \begin{array}{cc} a & b \ c & d \end{array} \right]\) is given by swapping the positions of \(a\) and \(d\), changing the signs of \(b\) and \(c\), and dividing each term by the determinant \((ad - bc)\). This results in the matrix:

• \(A^{-1} = \frac{1}{ad-bc}\left[ \begin{array}{cc} d & -b \ -c & a \end{array} \right]\)

Matrix Multiplication

Matrix multiplication is a methodical process and not as intuitive as scalar multiplication. When multiplying two matrices, the number of columns in the first matrix must match the number of rows in the second matrix. In the case of 2x2 matrices, the procedure involves the following steps:

• Select the first row of the first matrix and the first column of the second matrix.
• Multiply the corresponding elements and sum them to form the element of the product matrix in the first row and first column position.
• Repeat for every element, moving across rows in the first matrix and down columns in the second.

You end up constructing an entirely new matrix, called the product matrix, with elements based on these calculated sums.

For example, to validate if B is the inverse of A, we compute \(AB\) and \(BA\) and compare these results with the identity matrix.

Identity Matrix

The identity matrix acts as the neutral element in matrix multiplication, much like the number 1 in standard arithmetic. For any matrix, when multiplied by an identity matrix, it remains unchanged.

Identity matrices are square matrices where all elements on the main diagonal are ones, and all other elements are zeros.

• For a 2x2 identity matrix, it looks like this: \(I = \left[ \begin{array}{cc} 1 & 0 \ 0 & 1 \end{array} \right]\).

If a product of two matrices equals the identity matrix, it certifies that these two matrices are inversely related. For our exercise, calculating \(AB = I\) and \(BA = I\) proves that matrix B is indeed the inverse of matrix A.