Problem 25 - 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. $$ A D $$

Step-by-Step Solution

Answer

The multiplication \( AD \) cannot be performed because the matrices do not have compatible dimensions.

Step 1: Determine the Dimensions of Matrices

Matrix \( A \) is a 2x2 matrix and matrix \( D \) is a 1x2 matrix. To determine if we can multiply these matrices, we must ensure that the inner dimensions match.

Step 2: Check If Multiplication is Possible

For two matrices to be multiplied, the number of columns in the first matrix must be equal to the number of rows in the second matrix. In this case, matrix \( A \) has 2 columns and matrix \( D \) has 1 row. The inner dimensions (columns of \( A \) and rows of \( D \)) are not equal (2 ≠ 1), so multiplication is not possible.

Key Concepts

Matrix Dimensions

Matrix dimensions are crucial in linear algebra operations, particularly when dealing with matrix multiplication. A matrix with dimensions is typically described by the number of rows and columns it contains. For instance, matrix \( A \) is identified as a 2x2 matrix because it has 2 rows and 2 columns. Similarly, matrix \( D \) is a 1x2 matrix consisting of 1 row and 2 columns.

Understanding these dimensions is fundamental because it helps in determining if certain operations such as multiplication are feasible. Always denote matrices by their dimensions in the form \( m \times n \), where \( m \) represents the number of rows and \( n \) represents the number of columns. By consistently using this format, it becomes straightforward to assess compatibility between matrices in various algebraic operations.

Algebraic Operations

Algebraic operations involving matrices include addition, subtraction, and multiplication, among others. Each operation, however, has specific rules and requirements based on the matrices' dimensions.

For instance, in matrix addition and subtraction, the matrices involved must have identical dimensions. This means that each matrix must have the same number of rows and columns. The operations are then performed element-wise, resulting in a matrix of the same dimension. On the other hand, matrix multiplication involves a different set of rules where the inner dimensions play a pivotal role.

Understanding these rules is crucial for performing the correct operations and obtaining meaningful results from calculations involving matrices.

Multiplication Impossibility

Matrix multiplication is a defined operation only under certain conditions. The multiplication of two matrices \( A \) and \( D \) in the given exercise highlights a key concept—the impossibility due to incompatible dimensions.

When attempting to multiply matrices, the number of columns in the first matrix must equal the number of rows in the second matrix. This is a fundamental requirement for the multiplication to occur. In this scenario, matrix \( A \) has 2 columns, whereas matrix \( D \) only has 1 row, making their multiplication impossible. They do not satisfy the condition of equal inner dimensions, a core part of matrix multiplication theory.

Recognizing why multiplication can't proceed is just as vital as performing it, as it avoids unnecessary work and guides mathematical reasoning.

Inner Dimensions

The concept of inner dimensions is central to understanding why and how matrices can be multiplied. When two matrices are set for multiplication, the inner dimensions must match for the operation to occur.

To clarify, consider two matrices, \( A \) and \( B \). If \( A \) is an \( m \times n \) matrix and \( B \) is an \( n \times p \) matrix, multiplication is only possible because the 'inner' dimensions, \( n \), align perfectly. The resultant product will be an \( m \times p \) matrix.

Grasping this concept is not just about following mathematical convention; it ensures that calculations produce valid and useful data. In instances where inner dimensions clash, calculation must halt, reflecting an intrinsic check within matrix operations.