Problem 10 - Precalculus Mathematics for Calculus

Question

Perform the matrix operation, or if it is impossible, explain why. $$2\left[\begin{array}{lll} 1 & 1 & 0 \\ 1 & 0 & 1 \\ 0 & 1 & 1 \end{array}\right]+\left[\begin{array}{ll} 1 & 1 \\ 2 & 1 \\ 3 & 1 \end{array}\right]$$

Step-by-Step Solution

Answer

Matrix addition is not possible due to differing dimensions.

Step 1: Understand the Matrix Operation

The exercise requires performing an operation involving scalar multiplication of a matrix and the addition of another matrix. We need to first perform the scalar multiplication and then add the resulting matrix to another.

Step 2: Determine Matrix Dimensions

Check the dimensions of both matrices involved. The first matrix is a \(3 \times 3\) matrix, and the second matrix is a \(3 \times 2\) matrix. Matrix addition can only be performed on matrices with the same dimensions.

Step 3: Identify the Operation's Viability

Since the matrices have different dimensions, one being \(3 \times 3\) and the other \(3 \times 2\), they cannot be added together. Matrix addition requires both matrices to be of the same dimension.

Step 4: Conclusion Regarding Operation

Since the matrices have different dimensions, the matrix addition is impossible.

Key Concepts

Scalar Multiplication

Scalar multiplication is a straightforward yet essential operation when working with matrices. It involves multiplying every element of a given matrix by a constant, known as a scalar. For example, if we have a scalar value of 2 and a matrix, say \( A = \left[ \begin{array}{ccc} 1 & 2 & 3 \ 4 & 5 & 6 \ 7 & 8 & 9 \end{array} \right] \), then the result of this scalar multiplication is \( 2A = \left[ \begin{array}{ccc} 2 \times 1 & 2 \times 2 & 2 \times 3 \ 2 \times 4 & 2 \times 5 & 2 \times 6 \ 2 \times 7 & 2 \times 8 & 2 \times 9 \end{array} \right] \).

• Each element is multiplied by the scalar.
• The dimensions of the matrix are not affected by scalar multiplication.

Scalar multiplication is often a preliminary step before performing other operations, such as matrix addition or subtraction. In this exercise, multiplying the first matrix by 2 is the initial step before attempting addition with another matrix.

Matrix Addition

Matrix addition is a process by which two matrices of the same dimensions are added together. This means they must have the same number of rows and columns for the operation to be valid. In matrix addition, you add together corresponding elements from each matrix. Suppose we have matrices \( B = \left[ \begin{array}{cc} 1 & 2 \ 3 & 4 \end{array} \right] \) and \( C = \left[ \begin{array}{cc} 5 & 6 \ 7 & 8 \end{array} \right] \). The result is \( B + C = \left[ \begin{array}{cc} 1+5 & 2+6 \ 3+7 & 4+8 \end{array} \right] = \left[ \begin{array}{cc} 6 & 8 \ 10 & 12 \end{array} \right] \).

• Matrix addition can only occur if both matrices have identical dimensions.
• The resulting matrix will have the same dimensions as the original matrices.

If the dimensions do not match, as in the exercise problem, the addition is not possible. Recognizing matching dimensions is crucial to proceeding with an addition operation.

Matrix Dimensions

Understanding matrix dimensions is fundamental in matrix operations, as dimensions dictate which operations are feasible. A matrix's dimensions are described in terms of its rows and columns, denoted by \( m \times n \), where \( m \) is the number of rows and \( n \) the number of columns. For instance, a \( 2 \times 3 \) matrix has 2 rows and 3 columns.

• Matrix dimensions are vital in deciding operations like addition, subtraction, or multiplication.
• For example, in matrix addition, both matrices must have the same dimensions.

In our exercise, the dimensions of the two given matrices were\( 3 \times 3 \) and \( 3 \times 2 \). These differing dimensions make it impossible to add the matrices since their shapes do not align. Understanding and correctly identifying matrix dimensions ensures that mathematical operations on matrices are appropriately executed.