Problem 5 - Algebra and Trigonometry with Analytic Geometry

Question

Find, if possible, \(A+B, A-B, 2 A\), and \(-3 B\) $$ A=\left[\begin{array}{lll} 4 & -3 & 2 \end{array}\right], \quad B=\left[\begin{array}{lll} 7 & 0 & -5 \end{array}\right] $$

Step-by-Step Solution

Answer

\(A+B = \begin{bmatrix} 11 & -3 & -3 \end{bmatrix}\), \(A-B = \begin{bmatrix} -3 & -3 & 7 \end{bmatrix}\), \(2A = \begin{bmatrix} 8 & -6 & 4 \end{bmatrix}\), \(-3B = \begin{bmatrix} -21 & 0 & 15 \end{bmatrix}\)."

Step 1: Check the compatibility of matrices for addition and subtraction

For matrix addition and subtraction, both matrices must have the same dimensions. Here, both matrices \(A\) and \(B\) are 1x3, so we can proceed with addition and subtraction.

Step 2: Add matrices A and B

To add matrices \(A\) and \(B\), add their corresponding elements: \(A + B = \begin{bmatrix} 4+7 & -3+0 & 2+(-5) \end{bmatrix} = \begin{bmatrix} 11 & -3 & -3 \end{bmatrix}\).

Step 3: Subtract matrix B from A

To subtract \(B\) from \(A\), subtract corresponding elements of \(B\) from \(A\): \(A - B = \begin{bmatrix} 4-7 & -3-0 & 2-(-5) \end{bmatrix} = \begin{bmatrix} -3 & -3 & 7 \end{bmatrix}\).

Step 4: Multiply matrix A by scalar 2

To multiply each element of matrix \(A\) by 2, perform the operation: \(2A = \begin{bmatrix} 2\times4 & 2\times(-3) & 2\times2 \end{bmatrix} = \begin{bmatrix} 8 & -6 & 4 \end{bmatrix}\).

Step 5: Multiply matrix B by scalar -3

To multiply each element of matrix \(B\) by \(-3\), perform the operation: \(-3B = \begin{bmatrix} -3\times7 & -3\times0 & -3\times(-5) \end{bmatrix} = \begin{bmatrix} -21 & 0 & 15 \end{bmatrix}\).

Key Concepts

Matrix Addition

Matrix addition is a straightforward operation where we combine two matrices of the same dimensions by adding their corresponding elements. For example, if you have two 1x3 matrices like those in our exercise, the process becomes easy to understand. You can think of it like adding each number in a list with the corresponding number in another list.
To perform matrix addition:

• Ensure both matrices have the same dimension.
• Add corresponding elements together.

For instance, given matrices\[A = \begin{bmatrix} 4 & -3 & 2 \end{bmatrix}\]and\[B = \begin{bmatrix} 7 & 0 & -5 \end{bmatrix},\]their sum\(A + B\) becomes\[\begin{bmatrix} 4+7 & -3+0 & 2+(-5) \end{bmatrix} = \begin{bmatrix} 11 & -3 & -3 \end{bmatrix}.\]This result is also a 1x3 matrix, maintaining the same dimensions as the inputs.

Matrix Subtraction

Matrix subtraction is similar to matrix addition, but instead, we subtract elements of one matrix from the corresponding elements of the other. This operation requires that matrices be of the same dimensions. Like subtraction in arithmetic, matrix subtraction follows simple logic:

• Ensure both matrices have the same dimension.
• Subtract elements of one matrix from the corresponding elements of the other.

Using our example of matrices\[A = \begin{bmatrix} 4 & -3 & 2 \end{bmatrix}\]and\[B = \begin{bmatrix} 7 & 0 & -5 \end{bmatrix},\]the subtraction \(A - B\) is calculated as\[\begin{bmatrix} 4-7 & -3-0 & 2-(-5) \end{bmatrix} = \begin{bmatrix} -3 & -3 & 7 \end{bmatrix}.\]The resulting matrix still retains the 1x3 structure.

Scalar Multiplication

Scalar multiplication involves multiplying every entry in a matrix by a constant, or scalar. It is a basic yet powerful matrix operation used in various applications, like scaling matrices or adjusting their size. Here's how you can carry out scalar multiplication:

• Select a scalar value (a constant number).
• Multiply each element of the matrix by the scalar.

In our case, multiplying matrix\[A = \begin{bmatrix} 4 & -3 & 2 \end{bmatrix}\]by 2 gives:\[2A = \begin{bmatrix} 2\times4 & 2\times(-3) & 2\times2 \end{bmatrix} = \begin{bmatrix} 8 & -6 & 4 \end{bmatrix}.\]Similarly, multiplying matrix\[B = \begin{bmatrix} 7 & 0 & -5 \end{bmatrix}\]by \(-3\) results in:\[-3B = \begin{bmatrix} -3\times7 & -3\times0 & -3\times(-5) \end{bmatrix} = \begin{bmatrix} -21 & 0 & 15 \end{bmatrix}.\]With scalar multiplication, the matrix's dimensions remain unchanged.

1x3 Matrices

The matrices used in this example are 1x3 matrices, meaning they have one row and three columns. This specific dimension is quite simple yet serves as a great introduction to matrix operations without overwhelming complexity.
Understanding 1x3 matrices includes:

• Recognizing their structure: 1 row and 3 columns.
• Realizing they're often used in vector representation in three-dimensional space.
• Being aware that operations like addition, subtraction, and scalar multiplication are directly applicable.

Due to their straightforwardness, 1x3 matrices are perfect for illustrating basic matrix operations. They simplify learning as each operation directly maps one-to-one with easy-to-follow calculations, making it ideal for anyone new to matrices.