Problem 52 - College Algebra with Modeling and Visualization

Question

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

Step-by-Step Solution

Answer

Compute \( B^2 \), then \( 3A \), and subtract \( 3A \) from \( B^2 \).

Step 1: Calculate \( B^2 \)

To calculate \( B^2 \), we need to multiply matrix \( B \) by itself. The multiplication of two matrices \( B \) and \( B \) involves taking the dot product of rows of the first \( B \) and columns of the second \( B \). To find each element of the resulting matrix, multiply each corresponding element from the row and the column, then sum their products.

Step 2: Determine \( 3A \)

To find \( 3A \), multiply every element in matrix \( A \) by 3. This operation is called scalar multiplication. Apply this multiplication to all elements of matrix \( A \) element-wise.

Step 3: Subtract \( 3A \) from \( B^2 \)

Once matrices \( B^2 \) and \( 3A \) are obtained from the previous steps, subtract each element of \( 3A \) from the corresponding element in \( B^2 \) to find \( B^2 - 3A \). Make sure to align corresponding elements properly to get the correct result.

Key Concepts

Matrix Multiplication

Matrix multiplication is a fundamental operation in linear algebra. It involves the combination of two matrices to yield a third matrix, often revealing relationships across sets of linear equations.
To multiply two matrices, follow these basics: the number of columns in the first matrix must match the number of rows in the second. The resulting matrix will have dimensions determined by the number of rows of the first matrix and the number of columns of the second.
Start by taking the dot product of each row from the first matrix with each column of the second matrix. This means multiplying corresponding elements from the row and column and summing these products.

• Step 1: Identify compatible matrices.
• Step 2: Multiply corresponding elements.
• Step 3: Sum the products.

In the case of squaring matrix \( B \) as shown in the exercise, we take the matrix and multiply it by itself. This involves calculating each element of the resulting matrix by using the process described. Practice this operation often to get comfortable with the mechanics and to notice patterns in results.

Scalar Multiplication

Scalar multiplication is a simpler operation compared to matrix multiplication. Here, each component of a matrix is multiplied by a single number, known as a scalar. This operation scales the matrix without altering its structure.
Consider a scalar \( k \) and a matrix \( A \). To find \( kA \), multiply each element of \( A \) by \( k \). The resulting matrix will be the same size as the original matrix, but with each element scaled by the factor \( k \).
For example, multiplying matrix \( A \) by 3 in the given exercise is achieved by:

• Multiplying each entry of \( A \) by 3.
• Maintaining the structure and size of the original matrix.

This operation expands each value in the matrix uniformly, meaning all relative values within \( A \) remain consistent, just scaled by 3 in size.

Matrix Subtraction

Matrix subtraction is straightforward yet pivotal for advanced calculations. It involves element-wise subtraction of one matrix from another. To perform this operation, both matrices must be of the same dimensions.
Here's the step-by-step breakdown:

• Arrange both matrices in the same layout.
• Subtract the corresponding elements of one matrix from the other.
• Ensure that each matrix has identical dimensions to proceed.

This process is much like regular arithmetic subtraction but done simultaneously on two matrices.
In the exercise given, after calculating \( B^2 \) and \( 3A \), subtracting \( 3A \) from \( B^2 \) involves computing the difference of each corresponding element, thereby producing a resultant matrix. Matrix subtraction is used often in equations dealing with transformations and transitions, making it a core skill in handling linear equations and transformations.