Problem 21 - Precalculus: Mathematical for Calculus

Question

The matrices \(A, B, C, D, E, F,\) and \(G\) are defined as $$\begin{array}{l} A=\left[\begin{array}{rr} 2 & -5 \\ 0 & 7 \end{array}\right] \quad B=\left[\begin{array}{rrrr} 3 & \frac{1}{2} & 5 \\ 1 & -1 & 3 \end{array}\right] \quad C=\left[\begin{array}{rrr} 2 & -\frac{5}{2} & 0 \\ 0 & 2 & -3 \end{array}\right] \\ D=\left[\begin{array}{rrr} 7 & 3 \end{array}\right] & E=\left[\begin{array}{l} 1 \\ 2 \\ 0 \end{array}\right] \\ F=\left[\begin{array}{lll} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array}\right] \quad G=\left[\begin{array}{rrr} 5 & -3 & 10 \\ 6 & 1 & 0 \\ -5 & 2 & 2 \end{array}\right] \end{array}$$ Carry out the indicated algebraic operation, or explain why it cannot be performed. $$3 B+2 C$$

Step-by-Step Solution

Answer

The result of \(3B + 2C\) is \(\begin{bmatrix} 13 & -\frac{7}{2} & 15 \\ 3 & 1 & 3 \end{bmatrix}\).

Step 1: Matrix Multiplication Verification

We are tasked with finding the expression \(3B + 2C\). First, we check if our matrices can be added together. For two matrices to be added, they must have the same dimensions. Matrix \(B\) is a \(2 \times 3\) matrix and matrix \(C\) is also a \(2 \times 3\) matrix, so they are compatible for addition.

Step 2: Scalar Multiplication of Matrix B

Multiply each element of matrix \(B\) by the scalar 3:\[3B = 3\begin{bmatrix} 3 & \frac{1}{2} & 5 \ 1 & -1 & 3 \end{bmatrix} = \begin{bmatrix} 3 \times 3 & 3 \times \frac{1}{2} & 3 \times 5 \ 3 \times 1 & 3 \times -1 & 3 \times 3 \end{bmatrix} = \begin{bmatrix} 9 & \frac{3}{2} & 15 \ 3 & -3 & 9 \end{bmatrix}\]

Step 3: Scalar Multiplication of Matrix C

Multiply each element of matrix \(C\) by the scalar 2:\[2C = 2\begin{bmatrix} 2 & -\frac{5}{2} & 0 \ 0 & 2 & -3 \end{bmatrix} = \begin{bmatrix} 2 \times 2 & 2 \times -\frac{5}{2} & 2 \times 0 \ 2 \times 0 & 2 \times 2 & 2 \times -3 \end{bmatrix} = \begin{bmatrix} 4 & -5 & 0 \ 0 & 4 & -6 \end{bmatrix}\]

Step 4: Matrix Addition

Add the matrices \(3B\) and \(2C\):\[3B + 2C = \begin{bmatrix} 9 & \frac{3}{2} & 15 \ 3 & -3 & 9 \end{bmatrix} + \begin{bmatrix} 4 & -5 & 0 \ 0 & 4 & -6 \end{bmatrix} = \begin{bmatrix} 9+4 & \frac{3}{2} - 5 & 15+0 \ 3+0 & -3+4 & 9-6 \end{bmatrix} = \begin{bmatrix} 13 & -\frac{7}{2} & 15 \ 3 & 1 & 3 \end{bmatrix}\]

Key Concepts

Matrix Addition

Matrix addition is a straightforward operation between two matrices of the same size. Each element in the resulting matrix is the sum of the corresponding elements in the matrices being added. For instance, when adding two matrices like matrix \( B \) and matrix \( C \), both of which are \( 2 \times 3 \) matrices, you simply add each corresponding element from these matrices together.
To visualize, consider the element in the first row and first column of both matrices. The resulting matrix will have this element as the sum of the two elements from the input matrices. This operation looks like this for the first elements:

• If the first element of matrix \( B \) is 3 and that of matrix \( C \) is 2, then the first element of the resulting matrix is \( 3 + 2 = 5 \).

Matrix addition will only work if both matrices are of the same dimensions, otherwise, they are not compatible. Therefore, ensuring the matrices are compatible is a crucial step before adding them.

Scalar Multiplication

Scalar multiplication involves multiplying every element within a matrix by a single number, known as the 'scalar'. This operation expands or shrinks the matrix without changing its structure. For example, if we multiply matrix \( B \) by a scalar 3, each element within the matrix is multiplied by 3, turning a matrix such as \[B = \begin{bmatrix} 3 & \frac{1}{2} & 5 \ 1 & -1 & 3 \end{bmatrix}\] into \[3B = \begin{bmatrix} 9 & \frac{3}{2} & 15 \ 3 & -3 & 9 \end{bmatrix}\].
This operation is applied element-wise, meaning every element in the matrix is acted upon by the scalar individually, retaining the matrix's dimensions (a \( 2 \times 3 \) matrix remains \( 2 \times 3 \)).
Scalar multiplication is often a precursor to other operations, such as matrix addition, where you may need to first multiply each participating matrix by a scalar before adding.

Matrix Compatibility

Matrix compatibility refers to the conditions that must be met for certain operations, such as addition or multiplication, to be performed on matrices. For matrix addition, both matrices must have the same dimensions because each element must align with its counterpart in the other matrix. In simpler terms, if matrix \( A \) is a \( m \times n \) matrix, then matrix \( B \) must also be \( m \times n \) for them to be added.
In the case of scalar multiplication, compatibility is not a concern since scalars can operate on matrices of any size. However, for matrix multiplication, a different compatibility rule applies. Here, the number of columns in the first matrix must equal the number of rows in the second matrix.

• For example, a matrix with dimensions \( 2 \times 3 \) can only be multiplied by a matrix with dimensions \( 3 \times x \).

Understanding compatibility ensures that the operations can be executed correctly and that any attempt to perform illegal matrix operations is avoided.