Problem 24 - 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. $$D A$$

Step-by-Step Solution

Answer

The result of \( DA \) is \( \begin{bmatrix} 14 & -14 \end{bmatrix} \). Matrix multiplication is possible.

Step 1: Determine the Dimensions of D and A

Matrix \( D \) has dimensions \( 1 \times 2 \), since it has 1 row and 2 columns. Matrix \( A \) has dimensions \( 2 \times 2 \), since it has 2 rows and 2 columns.

Step 2: Check if Multiplication is Possible

For matrix multiplication \( DA \) to be possible, the number of columns in \( D \) must equal the number of rows in \( A \). Here, both matrices have a common dimension of 2, thus matrix \( DA \) is possible.

Step 3: Perform the Matrix Multiplication

Perform the matrix multiplication \( DA \) by finding the dot product between the row of \( D \) and each column of \( A \). This involves:\[ DA = \begin{bmatrix} 7 & 3 \end{bmatrix} \times \begin{bmatrix} 2 & -5 \ 0 & 7 \end{bmatrix} \]Calculating the elements: - First column: \( 7 \times 2 + 3 \times 0 = 14 + 0 = 14 \)- Second column: \( 7 \times (-5) + 3 \times 7 = -35 + 21 = -14 \)The resulting matrix is \( \begin{bmatrix} 14 & -14 \end{bmatrix} \).

Key Concepts

Matrix Dimensions

Understanding matrix dimensions is crucial in matrix operations. Every matrix has two dimensions: the number of rows and the number of columns. For instance, Matrix \( D \) is a \( 1 \times 2 \) matrix because it has 1 row and 2 columns.
Matrix \( A \) is a \( 2 \times 2 \) matrix, as it has 2 rows and 2 columns.

• Rows are counted horizontally and are the first dimension listed.
• Columns are counted vertically and are the second dimension listed.

When multiplying matrices, the order in which you arrange the matrices matters. Specifically, the number of columns of the first matrix must match the number of rows of the second matrix to perform multiplication.
If this condition is satisfied, you can proceed with the multiplication. Otherwise, the operation cannot be performed.

Dot Product

The dot product is an essential concept in matrix multiplication. When multiplying two matrices, you're essentially finding the dot product between rows of the first matrix and the columns of the second.
For example, in the multiplication \( DA \), Matrix \( D \) has the single row \([7, 3]\). Matrix \( A \) has two columns: \([2, 0]\) and \([-5, 7]\). During multiplication, each element of the row in \( D \) is multiplied by the corresponding element of a column in \( A \), and the results of these multiplications are then summed.

• First element: \( 7 \times 2 + 3 \times 0 = 14 + 0 = 14 \)
• Second element: \( 7 \times (-5) + 3 \times 7 = -35 + 21 = -14 \)

This results in a single row in the resulting matrix, highlighting the combination of numerical values in a meaningful way.

Matrix Algebra

Matrix algebra involves operations such as addition, subtraction, multiplication, and finding inverses with matrices.
Each of these operations has specific rules, primarily based on the dimensions of the matrices involved.

• Addition and Subtraction: Only possible when matrices have identical dimensions.
• Multiplication: Requires the inner dimensions to match, e.g., a \( m \times n \) matrix can be multiplied with an \( n \times p \) matrix.
• Inverse: Exists only for square matrices (where rows equal columns) and only if the determinant is non-zero.

Understanding these rules helps in solving complex problems efficiently as matrix algebra is a foundation for many applications in mathematics and applied sciences.

Precalculus

Precalculus serves as foundational groundwork that introduces key concepts for calculus, including matrices. Studying matrices in precalculus helps students understand structures that are ubiquitous in higher mathematics.
Matrix multiplication is one such fundamental topic.

• Matrices are used to represent linear transformations and systems of linear equations.
• Grasping the idea of matrix dimensions and products prepares students for vector spaces and linear mappings.

By understanding matrices during precalculus, students prepare for both theoretical and practical applications found in calculus and other advanced subjects. Matrices thus act as a stepping stone, solidifying comprehension of more abstract mathematical concepts.