Problem 38 - College Algebra

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

Step-by-Step Solution

Answer

\((AB)^2 = \begin{bmatrix} 640000 & 240000 \\ 0 & 640000 \end{bmatrix}\).

Step 1: Verify Matrix Multiplication Compatibility

To determine if matrices \(A\) and \(B\) can be multiplied, we must ensure that the number of columns in \(A\) is equal to the number of rows in \(B\). Both matrices \(A\) and \(B\) are 2x2 matrices, so their multiplication is possible.

Step 2: Compute \(AB\)

Multiply matrix \(A\) and matrix \(B\):\[A = \begin{bmatrix} -10 & 20 \ 5 & 25 \end{bmatrix}, \quad B = \begin{bmatrix} 40 & 10 \ -20 & 30 \end{bmatrix}\]To find \(AB\):\[AB = \begin{bmatrix} (-10)(40)+ (20)(-20) & (-10)(10) + (20)(30) \ (5)(40) + (25)(-20) & (5)(10) + (25)(30) \end{bmatrix}\]\[AB = \begin{bmatrix} -800 & 500 \ 0 & 800 \end{bmatrix}\]

Step 3: Compute \((AB)^2\)

Now that we have \(AB\), we need to find \((AB)\cdot(AB)\):\[AB = \begin{bmatrix} -800 & 500 \ 0 & 800 \end{bmatrix}\]Find \((AB)^2\):\[(AB)(AB) = \begin{bmatrix} (-800)(-800) + (500)(0) & (-800)(500) + (500)(800) \ (0)(-800) + (800)(0) & (0)(500) + (800)(800) \end{bmatrix}\]\[(AB)^2 = \begin{bmatrix} 640000 & 240000 \ 0 & 640000 \end{bmatrix}\]

Step 4: Verify Results and Provide Explanation

Check the calculations to ensure no arithmetic errors occurred and that each matrix was multiplied accurately. The operation of multiplying \((AB)\) by itself is valid, confirming that \((AB)^2\) is indeed a 2x2 matrix.

Key Concepts

Matrix Compatibility

In order to multiply two matrices together, it is essential to check their compatibility. This is known as matrix compatibility. For two matrices to be compatible for multiplication, the number of columns in the first matrix must be equal to the number of rows in the second matrix. This ensures that each element of the row vector in the first matrix can be paired with the corresponding element of the column vector in the second matrix.

For example, if matrix \(A\) is \(2 \times 2\) and matrix \(B\) is also \(2 \times 2\), these two matrices are compatible because the number of columns in \(A\) (which is 2) matches the number of rows in \(B\) (also 2). This allows us to proceed with the multiplication operation without any issues. Understanding this concept is crucial when dealing with more complex matrices and ensures that mathematical operations proceed smoothly.

Matrix Operations

Matrix operations involve various mathematical procedures such as addition, subtraction, and multiplication that are performed on matrices. Among these, matrix multiplication is one of the most widely used operations. It involves taking two matrices and combining them to produce a third matrix.

When we multiply two matrices, each element of the resulting matrix is calculated as the sum of the products of the corresponding elements of the row from the first matrix and the column from the second matrix. Continuing with our previous example, since matrices \(A\) and \(B\) are both \(2 \times 2\), the multiplication of \(A\) and \(B\) is performed by:

• Multiplying elements of the first row of \(A\) with the first column of \(B\) and summing them up for the first element.
• Repeating similar steps for the remaining elements, resulting in a new \(2 \times 2\) matrix \(AB\).

Performing this operation correctly involves careful attention to detail to ensure each element is computed accurately.

Matrix Arithmetic

Once compatibility is established and the basic operations are understood, matrix arithmetic, such as finding powers of matrices, becomes feasible. Matrix arithmetic extends beyond simple operations and includes operations like squaring a matrix.

Squaring a matrix involves multiplying the matrix by itself. This is denoted as \((AB)^2\), where you multiply matrix \(AB\) with itself. For the exercise at hand, starting with matrix \(AB\) from our calculations:

• Each element of the new matrix is found by taking the dot product of corresponding rows and columns of matrix \(AB\).
• Once these calculations are completed and errors are checked, you verify that the squared matrix retains the expected dimensions and properties.

By meticulously multiplying \(AB\) by itself, you arrive at \((AB)^2\), showcasing the power of matrix arithmetic in solving complex mathematical problems.