Problem 21 - Algebra and Trigonometry

Question

The matrices \(A, B, C, D, E, F,\) and \(G\) are defined as follows. $$A=\left[\begin{array}{rr}{2} & {-5} \\ {0} & {7}\end{array}\right] \quad B=\left[\begin{array}{rrr}{3} & {\frac{1}{2}} & {5} \\ {1} & {-1} & {3}\end{array}\right] \quad C=\left[\begin{array}{rrr}{2} & {-\frac{5}{2}} & {0} \\ {0} & {2} & {-3}\end{array}\right]$$ $$\begin{array}{l}{D=\left[\begin{array}{lll}{7} & {3}\end{array}\right]} & {E=\left[\begin{array}{lll}{1} \\ {1} \\ {2} \\ {0}\end{array}\right]} \\\ {F=\left[\begin{array}{lll}{1} & {0} & {0} \\ {0} & {1} & {0} \\ {0} & {0} & {1}\end{array}\right] \quad G=\left[\begin{array}{rrr}{5} & {-3} & {10} \\\ {6} & {1} & {0} \\ {-5} & {2} & {2}\end{array}\right]}\end{array}$$ 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: Understand Matrix Addition

To perform matrix addition, both matrices must have the same dimensions. Matrices are added by adding their corresponding elements.

Step 2: Understand Scalar Multiplication

Scalar multiplication involves multiplying every element of a matrix by a scalar value. For matrices \(3B\) and \(2C\), we will multiply each element of matrix \(B\) by 3 and each element of matrix \(C\) by 2.

Step 3: Scalar Multiplication for Matrix B

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

Step 4: Scalar Multiplication for Matrix C

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

Step 5: Add the Resulting Matrices

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

Key Concepts

Matrix Addition

Matrix addition is the process of adding two matrices by adding their corresponding elements. This operation can only be performed when both matrices involved have the same dimensions. This means that the number of rows and columns in the first matrix should match exactly with the number of rows and columns in the second matrix.
For instance, consider two matrices each having dimensions of 2x3 (2 rows and 3 columns), such as matrices A and B in a hypothetical example. Adding these matrices involves adding the elements in the first row and first column of both matrices, the first row and second column of both matrices, and so on. The result will be a new matrix of the same dimensions.

• For example, if we take a matrix with dimensions 2x3:

  Matrix 1: \[\begin{bmatrix}  1 & 2 & 3 \4 & 5 & 6\end{bmatrix}\]

  And another matrix:Matrix 2:\[\begin{bmatrix}  6 & 5 & 4 \3 & 2 & 1\end{bmatrix}\]

• The resulting matrix from their addition would be:

  Result: \[\begin{bmatrix}  1+6 & 2+5 & 3+4 \4+3 & 5+2 & 6+1  \end{bmatrix} =\begin{bmatrix}  7 & 7 & 7 \7 & 7 & 7\end{bmatrix}\]

This procedure is straightforward but requires precision, especially when dealing with larger matrices or when matrices contain fractions or decimal points.

Scalar Multiplication

Scalar multiplication is the process of multiplying every element within a matrix by a constant value, known as a "scalar." This operation is fundamental in matrix algebra and allows for the scaling of a matrix's entries by a consistent factor. The scalar, which can be any real number, essentially scales or dilates the magnitude of the original matrix without altering its dimensions.
For example, if you have a matrix \( M \):

\[M =\begin{bmatrix}  1 & 2 \3 & 4\end{bmatrix}\]

And you want to multiply it by a scalar, such as 3, the operation will yield:

\[3M = 3 	imes \begin{bmatrix}  1 & 2 \3 & 4\end{bmatrix} = \begin{bmatrix}  3 & 6 \9 & 12\end{bmatrix}\]

• Each element of the original matrix is simply multiplied by the scalar value.
• This alteration doesn't impact the structural dimensions of the matrix, merely the numerical values within it.

This method is often an intermediate step in more complex matrix operations like matrix-vector multiplication or solving systems of linear equations in algebra.

Matrix Dimensions

Matrix dimensions are a fundamental concept in understanding how matrices work. They define the size of a matrix and are expressed in terms of the number of rows and columns it has. Knowing the dimensions of matrices is crucial for performing operations like addition, subtraction, and multiplication, as these operations often have specific rules that relate to the sizes of the matrices involved.
A matrix is generally described as having a "m x n" dimension, where "m" refers to the number of rows and "n" refers to the number of columns. Understanding matrix dimensions allows you to swiftly determine whether matrix operations can be performed.

• For example, a matrix \[ A \]:

  A = \begin{bmatrix}1 & 2 & 3 \4 & 5 & 6 \end{bmatrix}

  is a 2x3 matrix because it has 2 rows and 3 columns.
• Similarly, consider a square matrix \[ B \]:

  B = \begin{bmatrix}7 & 8 \9 & 10 \end{bmatrix}

  This is a 2x2 matrix, indicating equal rows and columns.

If two matrices meet in size, they can be added or subtracted, and for multiplication, if two matrices have dimensions of "p x q" and "q x r," the result will be a "p x r" matrix. Understanding these concepts ensures proper application of matrix formulas and maximizes utilization of matrices in various fields like computer graphics, economics, and data analysis.