Problem 11 - Algebra 2

Question

Use matrices \(A, B, C,\) and \(D\) to find the following. $$A=\left[\begin{array}{cc}{2} & {3} \\ {5} & {6}\end{array}\right] \quad B=\left[\begin{array}{rr}{-1} & {7} \\ {0} & {-4}\end{array}\right] \quad C=\left[\begin{array}{rr}{9} & {-4} \\ {-6} & {5}\end{array}\right] \quad D=[2-5]$$ \(3 B-2 C\)

Step-by-Step Solution

Answer

\( 3B - 2C = \begin{bmatrix} -21 & 29 \\ 12 & -22 \end{bmatrix} \)

Step 1: Multiply Matrix C by 2

Matrix \(C\) is given by \(\begin{bmatrix}9 & -4 \ -6 & 5\end{bmatrix}\). Multiply each element of the matrix by 2, resulting in: \[2C = \begin{bmatrix}18 & -8 \ -12 & 10\end{bmatrix}\]

Step 2: Multiply Matrix B by 3

Matrix \(B\) is given by \(\begin{bmatrix}-1 & 7 \ 0 & -4 \end{bmatrix}\). Multiply each element of the matrix by 3, resulting in: \[3B = \begin{bmatrix} -3 & 21 \ 0 & -12 \end{bmatrix}\]

Step 3: Subtract 2C from 3B

With \(3B\) as \(\begin{bmatrix} -3 & 21 \ 0 & -12 \end{bmatrix}\) and \(2C\) as \(\begin{bmatrix}18 & -8 \ -12 & 10 \end{bmatrix}\), subtract \(2C\) from \(3B\) by subtracting corresponding elements to get:\[3B - 2C = \begin{bmatrix} -3-18 & 21-(-8) \ 0-(-12) & -12-10 \end{bmatrix} = \begin{bmatrix} -21 & 29 \ 12 & -22 \end{bmatrix} \]

Key Concepts

matrix multiplication

Matrix multiplication is a fundamental operation in linear algebra, allowing us to combine matrices in a specific way. Unlike regular multiplication, the process involves combining rows and columns of the matrices. When multiplying two matrices, say matrix \(A\) with size \((m\times n)\) and matrix \(B\) with size \((n\times p)\), the resulting matrix \(C\) will have the dimensions \((m\times p)\). Each element \(c_{ij}\) in the resulting matrix \(C\) is calculated as the sum of products of elements from the \(i\)-th row of \(A\) and the \(j\)-th column of \(B\).
Here's how it generally works:

• Align rows of the first matrix with the columns of the second matrix.
• Multiply corresponding elements and sum them to get the elements of the new matrix.

To perform matrix multiplication correctly, ensure that the number of columns in the first matrix is equal to the number of rows in the second matrix. If this condition is not met, the matrices cannot be multiplied.

matrix subtraction

Subtraction of matrices involves deducting corresponding elements of one matrix from another. The matrices involved in subtraction must be of the same size; otherwise, the operation is undefined. For matrices \(A\) and \(B\), the subtraction \(A - B\) yields a new matrix \(C\) where each element \(c_{ij}\) is calculated as \(a_{ij} - b_{ij}\).
To subtract two matrices:

• Ensure both matrices have the same dimensions.
• Subtract each element of the second matrix from the corresponding element of the first matrix.

This operation is straightforward and follows the same principles as regular numerical subtraction, except it's applied to every corresponding element in the matrices. In our exercise, matrix subtraction was the last step after performing scalar multiplications, showcasing how matrix operations can be combined sequentially.

scalar multiplication

Scalar multiplication is one of the simplest matrix operations. It involves multiplying every element of a matrix by a scalar (a single number). If you have a matrix \(A\) and a scalar \(k\), the resulting matrix is obtained by multiplying \(k\) with each element of \(A\).
Here are some basic steps involved in scalar multiplication:

• Take each element of the matrix.
• Multiply it by the scalar.

The application is straightforward but powerful when combined with other operations. For instance, it was used in the exercise to adjust matrices \(B\) and \(C\) individually with different scalars before performing further operations like subtraction. This adaptability is what makes scalar multiplication such an essential tool in algebra, facilitating transformations and changes in matrix values systematically.