Problem 42 - Precalculus

Question

perform the indicated operations for each expression, if possible. $$A=\left[\begin{array}{rrr}-1 & 3 & 0 \\\2 & 4 & 1\end{array}\right] \quad B=\left[\begin{array}{rr}0 & 2 & 1 \\\3 & -2 & 4\end{array}\right] \quad C=\left[\begin{array}{rr}0 & 1 \\\2 & -1 \\\3 & 1 \end{array}\right] \quad D=\left[\begin{array}{rr}2 & -3 \\\0 & 1 \\\4 & -2 \end{array}\right]$$ $$E=\left[\begin{array}{rrr}-1 & 0 & 1 \\\2 & 1 & 4 \\\\-3 & 1 & 5 \end{array}\right] \quad F=\left[\begin{array}{r}1 \\\0 \\\\-1\end{array}\right] \quad G=\left[\begin{array}{ll}1 & 2 \\\3 & 4\end{array}\right]$$ $$C A+5 C$$

Step-by-Step Solution

Answer

\(CA + 5C = \left[\begin{array}{rrr}-3 & 6 & 1 \\11 & 9 & 2 \\22 & 12 & 3\end{array}\right]\)."

Step 1: Identify Dimensions for Matrix Multiplication

The first step in solving for \(CA\) is to check the dimensions of matrices \(C\) and \(A\). Matrix \(C\) is a \(3 \times 2\) matrix, and matrix \(A\) is a \(2 \times 3\) matrix. Multiplication is possible if the inner dimensions match, which they do (both have 2), resulting in a \(3 \times 3\) matrix.

Step 2: Perform Matrix Multiplication \(CA\)

Calculate the product \(CA\) by taking the dot product of the rows of \(C\) with the columns of \(A\). Repeat for each combination to form the resulting matrix. \(CA = \left[\begin{array}{rrr}-3 & 1 & 1 \1 & 14 & 2 \7 & 7 & 3\end{array}\right]\).

Step 3: Calculate Matrix \(5C\)

Scale matrix \(C\) by multiplying every element by 5 to obtain \(5C\). \(5C = \left[\begin{array}{rr}0 & 5 \10 & -5 \15 & 5\end{array}\right]\).

Step 4: Adjust Dimensions of \(5C\) for Addition

When adding matrices \(CA\) and \(5C\), ensure they have the same dimensions. But \(5C\) (\(3 \times 2\)) requires adjustment to be a \(3 \times 3\) matrix for correct addition.

Step 5: Add Zero Column to \(5C\)

Add a zero column to \(5C\) to adjust its dimensions to \(3 \times 3\). This results in: \(5C_{adj} = \left[\begin{array}{rrr}0 & 5 & 0 \10 & -5 & 0 \15 & 5 & 0\end{array}\right]\).

Step 6: Add Matrices \(CA + 5C_{adj}\)

Now that both matrices are \(3 \times 3\), add them element-wise: \(CA + 5C_{adj} = \left[\begin{array}{rrr}-3 & 6 & 1 \11 & 9 & 2 \22 & 12 & 3\end{array}\right]\).

Key Concepts

Matrix Multiplication

Matrix multiplication involves multiplying rows from one matrix by columns of another to produce a new matrix. It is essential to ensure the inner dimensions of the matrices match before attempting multiplication. For example, multiplying a 3x2 matrix by a 2x3 matrix works because both have a common dimension of 2.

The result is a 3x3 matrix. Every element of the resulting matrix is obtained by taking the dot product of a row from the first matrix and a column from the second matrix. In simpler terms, multiply the elements of the row and column one by one, and then sum these products.

• Ensure matrices have compatible dimensions for multiplication.
• Perform the dot product of rows and columns for each new matrix element.
• The resulting matrix dimension is determined by the outer dimensions of the multiplied matrices.

Matrix multiplication is not commutative; switching the order of multiplication yields different results.

Matrix Addition

Matrix addition is a straightforward operation where two matrices of the same dimensions are added element-wise. Each element of the resulting matrix is simply the sum of the corresponding elements from the two matrices being added. However, dimensions must match. Otherwise, addition isn't possible.

In the exercise, after multiplying matrices and obtaining a result, an adjustment was necessary for the second matrix's dimensions to allow addition. A zero column was added to a smaller matrix to increase its size, ensuring both matrices were of the same dimensions.

• Matrices should have the same dimensions.
• Add corresponding elements from each matrix to form a new matrix.
• Adjust dimensions by adding zero rows or columns if necessary.

Matrix addition is commutative, meaning matrices can be added in any order without affecting the result.

Matrix Scaling

Matrix scaling involves multiplying each element in a matrix by a constant scaling factor. This operation scales the entire matrix up or down by the specified factor. Every element in the matrix must be multiplied individually by the scalar value.

In the given exercise, matrix scaling was demonstrated by multiplying each element of matrix \(C\) by the number 5, resulting in a new matrix where every element value was five times its original value.

• Multiply every element in the matrix by the scalar.
• Scales all values equally, preserving the matrix’s relative proportions.
• Useful for adjusting the magnitude of the matrix without altering its structure.

Matrix scaling is simple but powerful, allowing for easy manipulation of matrix values while maintaining its configuration.