Problem 1 - A Graphical Approach to Precalculus with Limits

Question

Determine whether A and B are imerses by calculating AB and BA. Do not use a calculator. $$A=\left[\begin{array}{ll} 5 & 7 \\ 2 & 3 \end{array}\right] ; B=\left[\begin{array}{rr} 3 & -7 \\ -2 & 5 \end{array}\right]$$

Step-by-Step Solution

Answer

Matrices A and B are inverses of each other since both AB and BA result in the identity matrix.

Step 1: Calculate AB

To determine if matrices \( A \) and \( B \) are inverses, we need to compute \( AB \). This involves multiplying matrix \( A \) by matrix \( B \):\[AB = \begin{bmatrix} 5 & 7 \ 2 & 3 \end{bmatrix} \cdot \begin{bmatrix} 3 & -7 \ -2 & 5 \end{bmatrix}\]Compute the elements of the resulting matrix:- First row, first column: \( (5 \cdot 3) + (7 \cdot -2) = 15 - 14 = 1 \)- First row, second column: \( (5 \cdot -7) + (7 \cdot 5) = -35 + 35 = 0 \)- Second row, first column: \( (2 \cdot 3) + (3 \cdot -2) = 6 - 6 = 0 \)- Second row, second column: \( (2 \cdot -7) + (3 \cdot 5) = -14 + 15 = 1 \)Thus, \( AB = \begin{bmatrix} 1 & 0 \ 0 & 1 \end{bmatrix} \), which is the identity matrix.

Step 2: Calculate BA

Now, compute \( BA \) to ensure it also results in the identity matrix:\[BA = \begin{bmatrix} 3 & -7 \ -2 & 5 \end{bmatrix} \cdot \begin{bmatrix} 5 & 7 \ 2 & 3 \end{bmatrix}\]Compute the elements of the resulting matrix:- First row, first column: \( (3 \cdot 5) + (-7 \cdot 2) = 15 - 14 = 1 \)- First row, second column: \( (3 \cdot 7) + (-7 \cdot 3) = 21 - 21 = 0 \)- Second row, first column: \( (-2 \cdot 5) + (5 \cdot 2) = -10 + 10 = 0 \)- Second row, second column: \( (-2 \cdot 7) + (5 \cdot 3) = -14 + 15 = 1 \)Thus, \( BA = \begin{bmatrix} 1 & 0 \ 0 & 1 \end{bmatrix} \), which is also the identity matrix.

Key Concepts

Matrix Multiplication

Matrix multiplication is a fundamental operation in linear algebra, and it allows us to combine matrices in a specific way. When we multiply two matrices, such as matrix \( A \) and matrix \( B \), we must ensure that the number of columns in the first matrix equals the number of rows in the second matrix. This ensures that the multiplication can be performed.

When multiplying, think of the first matrix as a series of rows and the second matrix as a series of columns. For each element in the resulting matrix:

• Take the corresponding row from the first matrix.
• Take the corresponding column from the second matrix.
• Multiply each pair of elements — one from the row and one from the column.
• Sum these products to get the element of the new matrix.

This process is repeated for every element in the resulting matrix, providing a new matrix that holds all these calculated values. Matrix multiplication is not commutative, meaning \( AB \) might not equal \( BA \). Hence, always follow the order strictly.

Identity Matrix

The identity matrix is a special type of matrix in linear algebra. It acts as a "one" in matrix arithmetic, much like the number 1 in regular arithmetic. The identity matrix is a square matrix with 1s on the diagonal (from top left to bottom right) and 0s elsewhere.

For example, the 2x2 identity matrix is:\[I = \begin{bmatrix} 1 & 0 \ 0 & 1 \end{bmatrix}\]Multiplying any matrix by the identity matrix retains the original matrix unchanged, just like multiplying a number by 1. This is true for both matrix multiplication orders: \( AI = A \) and \( IA = A \). In the context of inverses, if multiplying two matrices results in the identity matrix (e.g., \( AB = I \)), it indicates that these matrices are inverses of each other.

Checking if both \( AB \) and \( BA \) equal the identity matrix confirms that both matrices \( A \) and \( B \) are indeed inverses.

Determinants

Determinants are a unique and valuable property of square matrices. They are scalar values that can provide important insights into the matrix's properties, such as invertibility.

For a 2x2 matrix \( \begin{bmatrix} a & b \ c & d \end{bmatrix} \), the determinant is calculated as:\[\text{det} = ad - bc\]The determinant helps in determining whether a matrix has an inverse. A key rule is that a matrix is invertible if and only if its determinant is non-zero. If the determinant is zero, then the matrix does not have an inverse, and the row vectors in the matrix are linearly dependent.

For example, in determining whether a matrix is invertible and thus has an inverse, calculating its determinant gives us a clear answer. In practice, we perform matrix operations like finding inverses only on matrices with a non-zero determinant.