Problem 28 - 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. $$ G F $$

Step-by-Step Solution

Answer

Matrix multiplication of \( GF \) is possible and will yield a \( 3 \times 3 \) matrix.

Step 1: Identify Dimensions of Matrices

The matrix \( G \) is a \( 3 \times 3 \) matrix and the matrix \( F \) is also a \( 3 \times 3 \) matrix. To multiply two matrices, the number of columns in the first matrix must equal the number of rows in the second matrix. Here, \( G \) has 3 columns and \( F \) has 3 rows, so multiplication is possible.

Step 2: Perform Matrix Multiplication

To multiply matrix \( G \) by matrix \( F \), calculate each element of the resulting matrix as the dot product of the corresponding row of \( G \) and column of \( F \). The resulting matrix will also be a \( 3 \times 3 \) matrix since both \( G \) and \( F \) are \( 3 \times 3 \) matrices.

Key Concepts

Matrix Dimensions

Matrix dimensions refer to the number of rows and columns a matrix has. They are usually expressed as \( m \times n \) where \( m \) is the number of rows and \( n \) is the number of columns. To perform matrix multiplication, one must ensure that the dimensions of the matrices involved are compatible. Specifically, the number of columns in the first matrix must match the number of rows in the second matrix. For example, both matrices \( G \) and \( F \) have dimensions of \( 3 \times 3 \). This means they can be multiplied because the columns of \( G \) match the rows of \( F \). Whenever you are working with matrices, checking dimensions is a crucial first step. If this condition is not met, matrix multiplication cannot be performed.

Dot Product

The dot product is a key component in matrix multiplication. It involves multiplying corresponding elements from a row of the first matrix with a column of the second matrix and then summing those products. Consider a row vector \((a_1, a_2, a_3)\) from matrix \( G \) and a column vector \((b_1, b_2, b_3)\) from matrix \( F \). The dot product is computed as follows:

• Multiply each corresponding pair: \( a_1 \times b_1, a_2 \times b_2, a_3 \times b_3 \)
• Sum these products: \( a_1 \times b_1 + a_2 \times b_2 + a_3 \times b_3 \)

This result becomes an element in the resulting matrix. Understanding the dot product is essential to compute each element correctly in the resultant matrix.

Matrix Elements

Matrix elements are the individual values or entries within a matrix. In matrix multiplication, these elements are used extensively to calculate the resultant matrix. For instance, if you are multiplying matrices \(G\) and \(F\), you'd take elements from the rows of \(G\) and columns of \(F\), compute their dot products, and insert them in their respective positions in the new matrix. Each element in a matrix is typically denoted by \(a_{ij}\), where \(i\) and \(j\) represent its position based on the row and column, respectively. Understanding how each element is manipulated during multiplication helps in comprehending the overall matrix transformation that occurs.

Square Matrix

A square matrix is one with the same number of rows and columns. For example, both matrices \(G\) and \(F\) are square matrices with dimensions \(3 \times 3\). One important feature of square matrices is that they can be multiplied by themselves and other matrices of matching dimensions, such as with itself (a feature useful in many mathematical applications). Square matrices often appear in various mathematical contexts, such as systems of linear equations, where one might use matrix operations to find solutions. They also possess unique properties like having a determinant and potentially being invertible. Understanding square matrices is important because they simplify many mathematical operations and are more likely to have special properties compared to non-square matrices.