Problem 16 - Precalculus with Limits

Question

In Exercises 11-18, if possible, find (a) \(A+B\), (b) \(A-B\), (c ) \(3A\), and (d) \(3A-2B\). \(A = \left[ \begin{array}{r} -1 & 4 & 0 \\ 3 & -2 & 2 \\ 5 & 4 & -1 \\ 0 & 8 & -6 \\ -4 & -1 & 0 \end{array} \right]\), \(B = \left[ \begin{array}{r} -3 & 5 & 1 \\ 2 & -4 & -7 \\ 10 & -9 & -1 \\ 3 & 2 & -4 \\ 0 & 1 & -2 \end{array} \right]\)

Step-by-Step Solution

Answer

The result for each operation will be matrices of the same size (5x3) since all these operations are performed element by element on the compatible matrices.

Step 1: (a) Adding A and B

To find \(A + B\), add the corresponding elements of both matrices. The element in the first row and first column of the resultant matrix will be the sum of the first row and first column elements of the matrices A and B, and so forth for all the elements present in the matrices.

Step 2: (b) Subtracting B from A

For \(A - B\), perform element-wise subtraction; subtract each element of matrix B from the corresponding element of matrix A.

Step 3: (c) Multiplying A by 3

To find \(3A\), multiply each element of matrix A by 3. Scalar multiplication of a matrix involves multiplying every element of the matrix by the scalar.

Step 4: (d) Subtraction 2B from 3A

To compute \(3A - 2B\), first perform the scalar multiplication from part (c) for A, and then perform scalar multiplication for B by multiplying each of its elements by 2. Then subtract the resulting matrix from part (b) from matrix \(3A\).

Key Concepts

Matrix Addition

Matrix addition is simply adding corresponding elements from two matrices. Consider matrices A and B, each of which must be of the same dimensions. This means that if A is a 3x3 matrix, then B should also be a 3x3 matrix.

In matrix addition, one should:

• Identify corresponding elements. For example, add the element in the first row and first column of A to the element in the first row and first column of B.
• Continue adding these corresponding elements for all positions in the matrices.
• Ensure both matrices are aligned correctly based on their dimensions.

Matrix addition provides a way to combine matrices, summing up their individual values element by element.

Matrix Subtraction

Matrix subtraction involves subtracting each element of the second matrix from the corresponding element of the first matrix. Like addition, subtraction requires both matrices to have the same dimensions.

Here's how it works:

• Align the matrices so that each element corresponds correctly.
• Subtract the element in the first row and first column of matrix B from that of matrix A.
• Repeat this for all elements, proceeding row by row.

This process yields a new matrix that represents the difference between the two. In essence, matrix subtraction is about calculating the difference between each pair of structures from the matrices.

Scalar Multiplication

Scalar multiplication of a matrix involves multiplying every element of a matrix by a single number, called a scalar. This operation changes the size of each element according to the scalar value.

The process includes:

• Choosing a scalar, like the number 3 in the exercise.
• Multiplying each element of the matrix by this scalar value.
• Carry out the operation for all elements, maintaining their positions in the matrix.

This operation is basic yet powerful, enabling resizing of the matrix's numerical values without altering the matrix's structure.

Element-wise Operations

Element-wise operations involve performing calculations between matrices element by element. This can refer to addition, subtraction, multiplication, or division carried out individually between corresponding elements of matrices.

The steps are:

• Ensure that both matrices involved have identical dimensions, which is crucial for any element-wise operation.
• Carry out the intended operation (addition, subtraction, etc.) on each corresponding element.
• After operation, a resulting matrix emerges that reflects the combined values after the operation.

Such operations are crucial for linear algebra computations where element-by-element processing is required.