Problem 51 - College Algebra with Modeling and Visualization

Question

Use the given \(A\) and \(B\) to evaluate each expression. $$A=\left[\begin{array}{rrr}3 & -2 & 4 \\\5 & 2 & -3 \\\7 & 5 & 4\end{array}\right], B=\left[\begin{array}{rrr}1 & 1 & -5 \\\\-1 & 0 & -7 \\\\-6 & 4 & 3\end{array}\right]$$ $$3 A^{2}+2 B$$

Step-by-Step Solution

Answer

Perform matrix calculations as outlined to evaluate \(3A^2 + 2B\).

Step 1: Find the Square of Matrix A

First, multiply matrix \(A\) by itself to find \(A^2\). Compute each element in the resulting matrix by taking the dot product of the corresponding row from the first matrix \(A\) with the corresponding column from the second matrix \(A\). For example, the first element in the first row and column of \(A^2\) will be: \((3*3) + (-2*5) + (4*7) = 9 - 10 + 28 = 27.\)Repeat this process for each element in the resulting matrix.

Step 2: Multiply Matrix A² by 3

Once you have matrix \(A^2\), multiply each element of \(A^2\) by 3. Use standard scaler multiplication: each element in \(A^2\) is multiplied by 3.

Step 3: Multiply Matrix B by 2

Multiply matrix \(B\) by 2 in the same way: multiply each element of matrix \(B\) by 2.

Step 4: Add the Resulting Matrices

Take the two matrices obtained from Step 2 and Step 3 and add them together. To do this, add corresponding elements from each matrix.

Key Concepts

Matrix Multiplication

Matrix multiplication involves multiplying two matrices to obtain a new matrix. The process differs from multiplying regular numbers. This operation is only possible when the number of columns in the first matrix matches the number of rows in the second matrix.

Here's how you do it:

• Take a row from the first matrix and a column from the second matrix.
• Multiply corresponding components of the row and column together.
• Add the results of these multiplications to get an entry in the new matrix.

For example, the first element of the product matrix at the first row and first column is calculated by taking the first row of the first matrix, multiplying it by the first column of the second matrix, and summing the results. This method is repeated for each element of the resulting matrix.

In our exercise, for matrix \(A\), this involved a sequence of operations to square it, by multiplying it by itself, resulting in a matrix \(A^2\). This process required each element to be calculated based on the combination of rows and columns across the matrices.

Scalar Multiplication

Scalar multiplication is a simpler process than matrix multiplication. It involves multiplying every element of a matrix by the same scalar value. This process changes the magnitude of the matrix without altering its structure or the dimensions.

The operation is straightforward:

• Take each element of the matrix.
• Multiply it by the scalar value.
• Replace the original element with this new value.

In our exercise, after finding \(A^2\), each element in the resulting matrix was multiplied by 3. Similarly, each element of matrix \(B\) was multiplied by 2. The matrices remain the same size, but the numbers within them are scaled up or down by the scalar.

Matrix Addition

Matrix addition is the process of adding two matrices together to create a new matrix. To do this, both matrices must be the same size because you are adding corresponding elements.

Here's how to add matrices:

• Ensure both matrices have the same dimensions.
• Add each element in the first matrix to the corresponding element in the second matrix.
• The result is a new matrix of the same size with each entry being the sum of the corresponding entries of the two original matrices.

In this exercise, after performing scalar multiplication on both \(A^2\) and \(B\), the resulting matrices were added together. This involved directly adding the respective elements from each matrix to form the final solution matrix. This step combines the results of previous operations to achieve the final outcome.