Problem 3 - Precalculus: Functions and Graphs

Question

Find, if possible, \(A+B, A-B, 2 A,\) and \(-3 B\). $$A=\left[\begin{array}{rr} 6 & -1 \\ 2 & 0 \\ -3 & 4 \end{array}\right], \quad B=\left[\begin{array}{rr} 3 & 1 \\ -1 & 5 \\ 6 & 0 \end{array}\right]$$

Step-by-Step Solution

Answer

\(A + B = \left[\begin{array}{rr} 9 & 0 \\ 1 & 5 \\ 3 & 4 \end{array}\right], A - B = \left[\begin{array}{rr} 3 & -2 \\ 3 & -5 \\ -9 & 4 \end{array}\right], 2A = \left[\begin{array}{rr} 12 & -2 \\ 4 & 0 \\ -6 & 8 \end{array}\right], -3B = \left[\begin{array}{rr} -9 & -3 \\ 3 & -15 \\ -18 & 0 \end{array}\right]."

Step 1: Determine if A and B can be added or subtracted

To add or subtract matrices, they must have the same dimensions. Matrix \(A\) is a \(3 \times 2\) matrix, and matrix \(B\) is also \(3 \times 2\), so they can be added or subtracted.

Step 2: Calculate A + B

The sum of matrices \(A\) and \(B\) is found by adding corresponding elements.\[A + B = \left[\begin{array}{rr}6 + 3 & -1 + 1 \2 - 1 & 0 + 5 \-3 + 6 & 4 + 0\end{array}\right] = \left[\begin{array}{rr}9 & 0 \1 & 5 \3 & 4\end{array}\right]\]

Step 3: Calculate A - B

The difference of matrices \(A\) and \(B\) is found by subtracting corresponding elements.\[A - B = \left[\begin{array}{rr}6 - 3 & -1 - 1 \2 + 1 & 0 - 5 \-3 - 6 & 4 - 0\end{array}\right] = \left[\begin{array}{rr}3 & -2 \3 & -5 \-9 & 4\end{array}\right]\]

Step 4: Calculate 2A

To find \(2A\), multiply every element of \(A\) by 2.\[2A = 2 \times \left[\begin{array}{rr}6 & -1 \2 & 0 \-3 & 4\end{array}\right] = \left[\begin{array}{rr}12 & -2 \4 & 0 \-6 & 8\end{array}\right]\]

Step 5: Calculate -3B

To find \(-3B\), multiply every element of \(B\) by -3.\[-3B = -3 \times \left[\begin{array}{rr}3 & 1 \-1 & 5 \6 & 0\end{array}\right] = \left[\begin{array}{rr}-9 & -3 \3 & -15 \-18 & 0\end{array}\right]\]

Key Concepts

Matrix Addition

In matrix addition, we combine two matrices by adding their corresponding elements. For this operation to be possible, the matrices involved must have the same dimensions. This means that both should have the same number of rows and columns.
For instance, if matrix \(A\) and matrix \(B\) are both \(3 \times 2\) matrices, you can add them together because they have the same shape.

• The first element of \(A+B\) is the sum of the first elements of \(A\) and \(B\).
• Continue this process for every element in the matrices.

When performing the operations, ensure alignment of elements from each matrix is consistent and precise. This ensures that each operation is carried out correctly. When done right, matrix addition is straightforward and resembles regular arithmetic addition.

Matrix Subtraction

Matrix subtraction is the process of subtracting corresponding elements of one matrix from another. Like matrix addition, this operation can only be performed on matrices of the same dimensions.
To subtract matrix \(B\) from \(A\), represented as \(A-B\), follow these steps:

• Subtract each element of \(B\) from the corresponding element in \(A\), so the top left element of \(B\) is subtracted from the top left element of \(A\), and so forth.
• Continue until all corresponding elements have been subtracted.

The resulting matrix has the same dimensions and structure as the original matrices. Remember, matrix subtraction is straightforward, as long as you maintain the structure, noting each element's placement and order.

Scalar Multiplication

Scalar multiplication in matrices involves multiplying every element of a matrix by a given scalar, which is simply a regular number. This operation is versatile and does not require matrices of any specific size.
When performing scalar multiplication:

• Multiply each element in the matrix by the scalar number.
• The matrix's dimensions remain the same.

For example, if you have a matrix \(A\) and want to find \(2A\), multiply each element in \(A\) by 2. This applies similarly if one needs to find \(-3B\) for matrix \(B\). Scalar multiplication is essentially scaling the matrix by the scalar, making it a vital tool in many mathematical applications.

Matrix Dimensions

Matrix dimensions are crucial when performing matrix operations. The dimensions of a matrix are defined by the number of its rows and columns, noted as "rows \(\times\) columns".

• A \(3 \times 2\) matrix has 3 rows and 2 columns. Each position in this array holds a specific numerical value.
• When adding or subtracting matrices, their dimensions must match exactly.
• For scalar multiplication, the matrix dimensions can be of any size, as the operation impacts individual elements without concerning the matrix structure.

Understanding and identifying matrix dimensions at first glance simplifies decision-making regarding feasible operations. This knowledge helps avoid errors and streamlines calculations in matrix algebra.