Question

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

Short answer

\(-2A + 3A\) simplifies to \(A\).

Step-by-step solution

Understand the Expression

The given expression is \(-2A + 3A\). This expression involves scalar multiplication and addition of the vectors (or matrices) \(A\) and \(B\). The task is to simplify this expression.

Perform Scalar Multiplication

Multiply each element of matrix \(A\) by \(-2\) and by \(3\). For \(-2A\), multiply each element of \(A\) by \(-2\): \(-2 \times 3 = -6\) and \(-2 \times 5 = -10\), resulting in \(-2A = \begin{bmatrix} -6 \ -10 \end{bmatrix}\). For \(3A\), multiply each element of \(A\) by \(3\): \(3 \times 3 = 9\) and \(3 \times 5 = 15\), resulting in \(3A = \begin{bmatrix} 9 \ 15 \end{bmatrix}\).

Add the Resulting Matrices

Now, add the resulting matrices from Step 2, \(-2A\) and \(3A\). This means adding corresponding elements:\(-6 + 9 = 3\) and \(-10 + 15 = 5\). So the simplified matrix is \(\begin{bmatrix} 3 \ 5 \end{bmatrix}\).

Final Result

The expression \(-2A + 3A\) simplifies to \(\begin{bmatrix} 3 \ 5 \end{bmatrix}\). Thus, adding the scaled matrices returns the original matrix \(A\).

Key concepts

Scalar Multiplication

Scalar multiplication is a fundamental operation in matrix algebra. It involves multiplying each element of a matrix by a scalar, which is simply a real number. This operation changes the size of the matrix values but not its dimensions.
When performing scalar multiplication:

• Each element of the matrix gets multiplied by the scalar.
• The resulting matrix has the same dimensions as the original.

For instance, consider matrix \( A \):\[ A = \begin{bmatrix} 3 \ 5 \end{bmatrix} \]If you multiply \( A \) by \(-2\), you perform the operations \(-2 \times 3 = -6\) and \(-2 \times 5 = -10\), resulting in:\[ -2A = \begin{bmatrix} -6 \ -10 \end{bmatrix} \]
It’s the same for any scalar, whether positive or negative, as you only have to ensure each element is treated the same way, producing a consistent transformation across the matrix.

Matrix Addition

Matrix addition is another cornerstone of operations in matrix algebra. This operation is straightforward, but it requires matrices of the same dimensions to be combined.
The process of adding matrices involves:

• Adding corresponding elements together.
• The matrices must have the same number of rows and columns.

As seen in the exercise, consider two results from scalar multiplication, \(-2A\) and \(3A\), with:\[ -2A = \begin{bmatrix} -6 \ -10 \end{bmatrix} \quad\text{and}\quad 3A = \begin{bmatrix} 9 \ 15 \end{bmatrix} \]
Add them element-wise:

• \(-6 + 9 = 3\)
• \(-10 + 15 = 5\)

Thus, the sum of these matrices is:\[ \begin{bmatrix} 3 \ 5 \end{bmatrix} \]
Matrix addition essentially layers the changes of each contributing matrix to deliver a new result, assuming all operations behind each matrix are already completed and align in their broadest logical framework.

Matrix Simplification

Matrix simplification is often the culmination of multiple operations such as scalar multiplications and matrix additions. It's about reducing a complex expression into a more straightforward form, revealing the intrinsic properties or results of matrix operations.
In the given exercise, the expression \[ -2A + 3A \]was initially complex but was simplified through systematic application of matrix operations.

• First, execute scalar multiplications for both terms independently.
• Next, perform matrix addition to combine these results.

At the end of these steps, what you often find is a simplified matrix. In this case:\[ \begin{bmatrix} 3 \ 5 \end{bmatrix} \]
This particular simplification actually brought us back to matrix \( A \), teaching us that operations like these can sometimes reveal inherent relations or properties shared within the expression.