Problem 35 - College Algebra with Corequisite Support

Question

For the following exercises, use the matrices below to perform the indicated operation if possible. If not possible, explain why the operation cannot be performed. (Hint: \(A^{2}=A \cdot A\) ) \(A=\left[\begin{array}{rr}-10 & 20 \\ 5 & 25\end{array}\right], B=\left[\begin{array}{rr}40 & 10 \\ -20 & 30\end{array}\right], C=\left[\begin{array}{rr}-1 & 0 \\ 0 & -1 \\ 1 & 0\end{array}\right]\) \(B^{2}\)

Step-by-Step Solution

Answer

Matrix \( B^2 = \begin{bmatrix} 1400 & 700 \\ -1400 & 700 \end{bmatrix} \).

Step 1: Understand Matrix Squaring

The operation "squaring a matrix" implies multiplying the matrix by itself. For matrix \( B \), this means calculating \( B \cdot B \).

Step 2: Verify Matrix Multiplication Feasibility

Matrix multiplication \( B \cdot B \) is feasible if the number of columns in the first matrix \( B \) equals the number of rows in the second matrix \( B \). Since both are 2x2 matrices, the operation is possible.

Step 3: Perform the Matrix Multiplication

Calculate \( B^{2} = B \cdot B \). For matrices \( B = \begin{bmatrix} 40 & 10 \ -20 & 30 \end{bmatrix} \), multiply each element accordingly:\[B^{2} = \begin{bmatrix} 40 & 10 \ -20 & 30 \end{bmatrix} \times \begin{bmatrix} 40 & 10 \ -20 & 30 \end{bmatrix} = \begin{bmatrix} (40\times40 + 10\times(-20)) & (40\times10 + 10\times30) \ (-20\times40 + 30\times(-20)) & (-20\times10 + 30\times30) \end{bmatrix}\]

Step 4: Simplify the Matrix Elements

Calculate each element of the resulting 2x2 matrix:1. Top-left element: \( 40\times40 + 10\times(-20) = 1600 - 200 = 1400 \)2. Top-right element: \( 40\times10 + 10\times30 = 400 + 300 = 700 \)3. Bottom-left element: \( -20\times40 + 30\times(-20) = -800 - 600 = -1400 \)4. Bottom-right element: \( -20\times10 + 30\times30 = -200 + 900 = 700 \)Thus, \( B^{2} = \begin{bmatrix} 1400 & 700 \ -1400 & 700 \end{bmatrix} \).

Key Concepts

Matrix Multiplication

Matrix multiplication is a fundamental operation in matrix algebra. It involves two matrices where the number of columns in the first matrix must equal the number of rows in the second matrix. This is the compatibility rule for multiplication.

This operation is performed by taking the dot product of rows from the first matrix with columns from the second matrix. To break it down further:

• Consider two matrices, say Matrix A with dimensions (n x m) and Matrix B with dimensions (m x p).
• The resulting matrix from multiplying A with B will be a new matrix with dimensions (n x p).

Each element in the resulting matrix is computed by summing the products of corresponding elements. This is a critical step, as small errors can lead to incorrect results.

Matrix multiplication is non-commutative, meaning that multiplying matrix A by B does not necessarily give the same answer as multiplying B by A.

Matrix Squaring

Matrix squaring is a specific case of matrix multiplication where a single matrix is multiplied by itself. The operation is denoted as \(A^2 = A \cdot A\). This implies the matrix must be a square matrix, meaning it possesses the same number of rows and columns.

How can we check if a matrix can be squared?

• First, verify the dimensions of the matrix—only square matrices can be squared.
• Next, ensure the number of columns equals the number of rows. This is necessary for the matrix to be compatible for multiplication with itself.

On performing matrix squaring, it involves multiplying each row with respective columns as discussed in matrix multiplication. Despite seeming straightforward, this operation can significantly affect the characteristics of the matrix, such as eigenvalues, which may lead to applications in various fields like physics and statistics.

2x2 Matrices

2x2 matrices are the simplest form of square matrices, consisting of 2 rows and 2 columns. They are often introduced when learning matrix operations because of their simplicity.

Why focus on 2x2 matrices?

• They make the perfect example before moving on to larger matrices.
• The computations are less complex compared to larger matrices, which makes them ideal for educational purposes.
• The patterns and results obtained from operations like squaring and inversion in 2x2 matrices pave the way for understanding larger matrices.

In the context of squaring a 2x2 matrix, the calculation is relatively straightforward. For example, given a matrix \(B = \begin{bmatrix} a & b \ c & d \end{bmatrix}\) the squared matrix \(B^2\) is calculated by multiplying B by itself. Despite their simplicity, 2x2 matrices allow learners to grasp and visualize complex mathematical concepts effectively.