Problem 38 - 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: \(A^{2}=A \cdot A\) ) \(A=\left[\begin{array}{rr}-10 & 20 \\ 5 & 25\end{array}\right], B=\left[\begin{array}{rr}40 & 10 \\ -20 & 30\end{array}\right], C=\left[\begin{array}{rr}-1 & 0 \\ 0 & -1 \\ 1 & 0\end{array}\right]\) \(A^{2} B^{2}\)

Step-by-Step Solution

Answer

The matrix operation is possible, and the result of \(A^2B^2\) is \(\begin{bmatrix} -140000 & 350000 \\ -910000 & 560000 \end{bmatrix}\).

Step 1: Verify the Dimensions of Matrix Multiplication

Matrix multiplication can only occur if the number of columns in the first matrix equals the number of rows in the second matrix. Matrix \(A\) is \(2 \times 2\) and multiplying it by itself is possible because the number of columns (2) matches the number of rows (2). Similarly, \(B\) is also \(2 \times 2\), thus \(B^2\) is possible. However, after calculating \(A^2\) and \(B^2\), they both result in \(2 \times 2\) matrices, so \(A^{2}B^{2}\) is possible.

Step 2: Calculate A Squared \(A^{2}\)

To find \(A^{2}\), compute \(A \cdot A\) using matrix multiplication. Calculate the entries:\[A^{2} = \begin{bmatrix} -10 & 20 \ 5 & 25 \end{bmatrix} \cdot \begin{bmatrix} -10 & 20 \ 5 & 25 \end{bmatrix} = \begin{bmatrix} (-10)(-10) + (20)(5) & (-10)(20) + (20)(25) \ (5)(-10) + (25)(5) & (5)(20) + (25)(25) \end{bmatrix}\]\[= \begin{bmatrix} 100 + 100 & -200 + 500 \ -50 + 125 & 100 + 625 \end{bmatrix} = \begin{bmatrix} 200 & 300 \ 75 & 725 \end{bmatrix}\]

Step 3: Calculate B Squared \(B^{2}\)

To find \(B^{2}\), compute \(B \cdot B\):\[B^{2} = \begin{bmatrix} 40 & 10 \ -20 & 30 \end{bmatrix} \cdot \begin{bmatrix} 40 & 10 \ -20 & 30 \end{bmatrix} = \begin{bmatrix} (40)(40) + (10)(-20) & (40)(10) + (10)(30) \ (-20)(40) + (30)(-20) & (-20)(10) + (30)(30) \end{bmatrix}\]\[= \begin{bmatrix} 1600 - 200 & 400 + 300 \ -800 - 600 & -200 + 900 \end{bmatrix} = \begin{bmatrix} 1400 & 700 \ -1400 & 700 \end{bmatrix}\]

Step 4: Calculate the Product \(A^{2}B^{2}\)

Now compute \(A^{2} \cdot B^{2}\):\[A^{2}B^{2} = \begin{bmatrix} 200 & 300 \ 75 & 725 \end{bmatrix} \cdot \begin{bmatrix} 1400 & 700 \ -1400 & 700 \end{bmatrix}\]\[= \begin{bmatrix} (200)(1400) + (300)(-1400) & (200)(700) + (300)(700) \ (75)(1400) + (725)(-1400) & (75)(700) + (725)(700) \end{bmatrix}\]\[= \begin{bmatrix} 280000 - 420000 & 140000 + 210000 \ 105000 - 1015000 & 52500 + 507500 \end{bmatrix} = \begin{bmatrix} -140000 & 350000 \ -910000 & 560000 \end{bmatrix}\]

Step 5: Confirm the Solution

The final product \(A^{2}B^{2}\) after calculations is:\[A^{2}B^{2} = \begin{bmatrix} -140000 & 350000 \ -910000 & 560000 \end{bmatrix}\]This confirms that the matrix product is correctly computed as all dimensions and calculations align.

Key Concepts

Matrix Operations

Matrix operations are fundamental in linear algebra and involve processes such as addition, subtraction, and multiplication. While addition and subtraction require matrices to have the same dimensions, matrix multiplication is a bit more complex.

• To multiply two matrices, the number of columns in the first matrix must match the number of rows in the second matrix.
• This ensures the calculated product has meaningful values and valid dimensions.

Matrix multiplication is not commutative, which means that changing the order of the matrices being multiplied can lead to different results. This non-commutative nature requires careful attention to the order of matrices whenever multiplication is performed.
Additionally, for any matrix, multiplying by an identity matrix (which has ones on the diagonal and zeros elsewhere) will yield the original matrix. Matrix operations provide the tools for solving systems of equations, transformations, and many more applications in mathematics and engineering.

Matrix Dimensions

Matrix dimensions are defined as "rows by columns," written as \( m \times n \) where \( m \) is the number of rows and \( n \) is the number of columns. Understanding matrix dimensions is crucial to performing operations.

• If a matrix has dimensions \( 2 \times 2 \), it means there are 2 rows and 2 columns.
• When multiplying matrices, as mentioned, the inner dimensions must align; this means the number of columns in the first matrix must be equal to the number of rows in the second matrix.
• The resulting product will have dimensions from the outer dimensions of the multiplied matrices.

For instance, multiplying two \( 2 \times 2 \) matrices results in another \( 2 \times 2 \) matrix. Comprehending matrix dimensions helps determine whether an operation is possible and aids in predicting the shape of the resultant matrix. This can prevent errors in calculations due to incompatible matrices.

Square of a Matrix

The square of a matrix, denoted \( A^2 \), is obtained by multiplying the matrix by itself. This operation is only possible for square matrices, which are matrices where the number of rows equals the number of columns, as their dimensions are \( n \times n \).

• The result of squaring a matrix is also a matrix of the same dimensions.
• Squaring is different from adding a matrix to itself, as each element is the result of the dot product of rows and columns, not mere addition.

For example, squaring a matrix \( A \) that is \( 2 \times 2 \) involves performing matrix multiplication: \( A \times A \). Each entry of the resulting matrix is computed based on sums of products of corresponding elements.
The square of a matrix has applications in computing powers of matrices, especially in mathematical models such as Markov chains, which rely heavily on matrix operations. Understanding the square of a matrix is an essential aspect of working with matrices in various scientific computations.

Matrix Product Computation

Computing the product of matrices involves a sequence of multiplications and additions between elements of the matrices. Given matrices \( A \) and \( B \), their matrix product \( AB \) is obtained by computing a series of dot products.

• In each entry of the resulting matrix product, corresponding elements from the rows of the first matrix and columns of the second matrix are multiplied and summed up.
• The computation follows the formula where the entry in the i-th row and j-th column is \( c_{ij} = \sum_{k=1}^{n} a_{ik} b_{kj} \).

This means every row interacts with every column for the multiplication. For example, if \( A \) and \( B \) are both \( 2 \times 2 \) matrices, then their product is calculated as:

\[ AB = \begin{bmatrix} a_{11} \, & a_{12} \ a_{21} \, & a_{22} \end{bmatrix} \cdot \begin{bmatrix} b_{11} \, & b_{12} \ b_{21} \, & b_{22} \end{bmatrix} = \begin{bmatrix}a_{11}b_{11} + a_{12}b_{21} \, & a_{11}b_{12} + a_{12}b_{22} \ a_{21}b_{11} + a_{22}b_{21} \, & a_{21}b_{12} + a_{22}b_{22} \end{bmatrix} \]

Matrix product computation is an algorithmic process that requires precision in calculation as small errors can propagate through subsequent mathematical operations. It is foundational for solving various practical problems and is extensively used in computer graphics and physics simulations.