Question

Matrices \(A\) and \(B\) are given below. Simplify the given expression. $$ A=\left[\begin{array}{l} 3 \\ 5 \end{array}\right] \quad B=\left[\begin{array}{c} -2 \\ 4 \end{array}\right] $$ $$ 4 B-2 A $$

Short answer

The simplified expression is \( \begin{bmatrix} -14 \\ 6 \end{bmatrix} \).

Step-by-step solution

Multiplying Matrix B by Scalar

Multiply each element of matrix \( B \) by the scalar 4. The matrix \( B \) is \( \begin{bmatrix} -2 \ 4 \end{bmatrix} \).\[ 4 \times B = 4 \times \begin{bmatrix} -2 \ 4 \end{bmatrix} = \begin{bmatrix} 4 \times -2 \ 4 \times 4 \end{bmatrix} = \begin{bmatrix} -8 \ 16 \end{bmatrix} \]

Multiplying Matrix A by Scalar

Multiply each element of matrix \( A \) by the scalar 2. The matrix \( A \) is \( \begin{bmatrix} 3 \ 5 \end{bmatrix} \).\[ 2 \times A = 2 \times \begin{bmatrix} 3 \ 5 \end{bmatrix} = \begin{bmatrix} 2 \times 3 \ 2 \times 5 \end{bmatrix} = \begin{bmatrix} 6 \ 10 \end{bmatrix} \]

Subtracting Matrices

Subtract matrix \( 2A \) from matrix \( 4B \). Perform element-wise subtraction.\[ 4B - 2A = \begin{bmatrix} -8 \ 16 \end{bmatrix} - \begin{bmatrix} 6 \ 10 \end{bmatrix} = \begin{bmatrix} -8 - 6 \ 16 - 10 \end{bmatrix} = \begin{bmatrix} -14 \ 6 \end{bmatrix} \]

Key concepts

Scalar Multiplication

In matrix algebra, scalar multiplication is a very straightforward operation. It involves multiplying every entry of a matrix by a single number, known as a scalar. The idea here is simple and doesn't change the structure of the matrix; it just scales its elements.
Let's illustrate this by revisiting the expression \(4B\) from the exercise:

• Take each element of matrix \(B\) and multiply by 4.
• The result of \(4 imes B\) is obtained by computing each entry as \(4 imes -2 = -8\) and \(4 imes 4 = 16\).

This same principle applies when multiplying matrix \(A\) by the scalar 2.
Just take each element of matrix \(A\) and scale it by 2, resulting in \(2A = \begin{bmatrix} 6 \ 10 \end{bmatrix}\). This process of multiplication is crucial in altering matrix values without changing their arrangement in rows or columns.

Matrix Subtraction

Matrix subtraction is an arithmetic operation where you subtract corresponding elements of two matrices to form a new matrix. Make sure both matrices are of the same dimensions—this is essential for subtraction. Here’s how it works, using matrices \(4B\) and \(2A\) from the exercise:

• Element-wise subtraction means subtracting each element of matrix \(2A\) from the corresponding element in \(4B\).
• For instance, in our current matrices, subtract \(6\) from \(-8\) and \(10\) from \(16\) to get \(-14\) and \(6\) respectively.

The result is a new matrix \(\begin{bmatrix} -14 \ 6 \end{bmatrix}\). This operation forms the backbone of matrix expression simplifications, where element alignment ensures accurate solving.

Elementary Row Operations

While elementary row operations are not a direct part of this specific exercise, they are worth mentioning as they form a key component in matrix manipulation useful in more complex problems.
These operations include:

• Row switching — swapping two rows within a matrix.
• Row multiplication — multiplying all entries in a row by a non-zero scalar.
• Row addition — adding or subtracting a multiple of one row to another.

This set of operations is vital for transforming matrices into desired forms, such as row-echelon form, which is used in solving systems of linear equations.
Understanding these operations builds a solid foundation for more advanced matrix operations and computational linear algebra tasks.