Problem 39 - College Algebra

Question

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

Step-by-Step Solution

Answer

The operation \((AB)^2\) results in \(\begin{bmatrix} 490000 & 0 \\ 0 & 490000 \end{bmatrix}\).

Step 1: Verify Compatibility for Matrix Multiplication

To perform the operation \((AB)^{2}\), we first need to check if matrices \(A\) and \(B\) can be multiplied. Matrix \(A\) is a \(2 \times 2\) matrix, and matrix \(B\) is also a \(2 \times 2\) matrix. For multiplication, the number of columns in the first matrix must equal the number of rows in the second matrix. Since both matrices have dimensions \(2 \times 2\), they can be multiplied.

Step 2: Perform Matrix Multiplication AB

Multiply matrices \(A\) and \(B\):\[AB = \begin{bmatrix}-10 & 20 \ 5 & 25 \end{bmatrix} \cdot \begin{bmatrix}40 & 10 \ -20 & 30 \end{bmatrix} \]Calculate each element of the resulting matrix:- First row, first column: \((-10)(40) + (20)(-20) = -400 - 400 = -800\)- First row, second column: \((-10)(10) + (20)(30) = -100 + 600 = 500\)- Second row, first column: \((5)(40) + (25)(-20) = 200 - 500 = -300\)- Second row, second column: \((5)(10) + (25)(30) = 50 + 750 = 800\)Thus, \(AB = \begin{bmatrix} -800 & 500 \ -300 & 800 \end{bmatrix}\).

Step 3: Square the Resulting Matrix

Now that we have \(AB\), which is \(\begin{bmatrix} -800 & 500 \ -300 & 800 \end{bmatrix}\), we need to calculate \((AB)^2 = (AB) \times (AB)\).Multiply \(AB\) by itself:- First row, first column: \((-800)(-800) + (500)(-300) = 640000 - 150000 = 490000\)- First row, second column: \((-800)(500) + (500)(800) = -400000 + 400000 = 0\)- Second row, first column: \((-300)(-800) + (800)(-300) = 240000 - 240000 = 0\)- Second row, second column: \((-300)(500) + (800)(800) = -150000 + 640000 = 490000\)Thus, \((AB)^2 = \begin{bmatrix} 490000 & 0 \ 0 & 490000 \end{bmatrix}\).

Key Concepts

Matrix Dimensions

When discussing matrices, one fundamental concept to understand is their dimensions. The dimensions of a matrix are determined by the number of rows and columns it has. For example, a matrix with 2 rows and 3 columns is referred to as a \(2 \times 3\) matrix. Understanding the dimensions of matrices is crucial for performing operations like addition, subtraction, and especially multiplication.

Matrix multiplication requires strict adherence to a rule regarding dimensions: the number of columns in the first matrix must match the number of rows in the second matrix. For instance, if matrix \(A\) is a \(2 \times 2\) and matrix \(B\) is also a \(2 \times 2\), they are compatible for multiplication because the number of columns in \(A\) equals the number of rows in \(B\).

However, a \(2 \times 3\) matrix can only be multiplied with a \(3 \times \text{n}\) matrix, where \(\text{n}\) is any positive integer indicating the number of columns in the second matrix. This makes understanding matrix dimensions foundational for correctly performing matrix operations.

Square of a Matrix

Squaring a matrix might initially seem a bit mysterious, but it's grounded in basic multiplication. Squaring a matrix refers to multiplying a matrix by itself. In our exercise, we are finding \((AB)^2\), which means calculating \((AB) \times (AB)\).

To understand this, imagine squaring a simple number, say 3, which means multiplying 3 by itself, resulting in 9. For matrices, it's quite similar conceptually but involves more calculations.

• If \(A\) is a square matrix (like a \(2 \times 2\)), then \(A^2\) represents \(A \times A\).
• Operations must adhere to matrix multiplication rules, considering rows and columns during multiplication.

This process becomes a bit more intricate with matrices because each element of the resulting matrix is computed by summing the products of corresponding row and column elements. Hence, squaring a matrix amplifies the challenge, requiring precise evaluation of multiple terms in the resulting matrix.

Matrix Product Verification

Verifying the result of a matrix product operation is essential to ensure the calculations are performed accurately. After performing an initial matrix multiplication, like \(AB\) in our exercise, the resulting matrix must be checked for correctness before proceeding to further operations like squaring.

This verification process involves examining each element of the resulting matrix by recalculating their contributions:

• First row, first column: Ensure the sum of products from the first row of the first matrix and first column of the second matrix is correct.
• Continuing in the same manner for all elements, confirm the calculations respect order and dimension rules.

For instance, if during calculation, one finds discrepancies, it may indicate missteps in multiplication rules or basic arithmetic. In our scenario, reviewing the elements: \([-800, 0]\) and \([0, 490000]\), of \((AB)^2\), ensures correctness. Scrutiny at each step fortifies understanding and provides a safety net against errors, reinforcing confidence in the final outcome.