Problem 38 - 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. $$ A B E $$

Step-by-Step Solution

Answer

The operation \( ABE \) cannot be performed due to incompatible matrix dimensions.

Step 1: Determine Dimensions for Multiplication

The matrices involved in the expression \( ABE \) are \( A : 2 \times 2 \), \( B : 2 \times 3 \), and \( E : 4 \times 1 \). We need to determine whether these matrices can be multiplied in sequence.

Step 2: Validate Multiplication Sequence

The multiplication can be performed if the number of columns in the first matrix matches the number of rows in the second matrix. Start by checking \( A \cdot B \): Since the number of columns in \( A (2) \) is equal to the number of rows in \( B (2) \), \( AB \) is possible. \( AB \) will be a \(2 \times 3\) matrix.

Step 3: Check Compatibility for Further Multiplication

After multiplying \( A \) by \( B \), the resulting matrix is \( 2 \times 3 \). Attempting to multiply this by \( E \) (which is \( 4 \times 1 \)) requires the number of columns in the resultant matrix \( 2 \times 3 \) to match the number of rows in \( E (4) \). Since \( 3 eq 4 \), multiplication \( (AB)E \) is not possible.

Key Concepts

Matrix Dimensions

Every matrix has dimensions, represented by row and column counts, such as a **2 x 3** matrix having 2 rows and 3 columns. Dimensions are crucial clues in the world of matrices.
Understanding the size of a matrix is a key step to deciding on further operations. We denote dimensions like this:

• In matrix A, labeled as a **2 x 2**, it indicates 2 rows and 2 columns each holding specific entries.
• Matrix B is a **2 x 3** matrix, meaning 2 rows extend over 3 columns.
• Matrix E stands out as a **4 x 1** or a column matrix.

It's like knowing how big and in what shape blocks you'll use in a toy building set. Knowing matrix dimensions is essential when considering operations like addition, subtraction, and especially multiplication.

Matrix Compatibility

Matrix compatibility is about whether matrices can be interacted with in operations, especially multiplication. To multiply matrices, you must have compatible dimensions. For example:

• When multiplying two matrices, the number of columns in the first matrix must equal the number of rows in the second.
• Taking matrix A **(2 x 2)** and B **(2 x 3):** you can multiply A by B since 2 columns in A equals 2 rows in B, resulting in a new **(2 x 3)** matrix.
• For further multiplying by E **(4 x 1),** which has 4 rows, you require the result of AB **(2 x 3)** to have 3 rows. Here, the incompatibility occurs, blocking the operation.

To visualize, think about the ends of two Lego bricks needing to perfectly match to form a new structure; every peg (number of columns) should fit into every corresponding hole (number of rows) of the next piece for successful assembly.

Matrix Algebra

Matrix algebra involves performing mathematical operations on matrices, akin to algebra with numbers, but following rules concerning their dimensions.
With matrices, operations available include addition, subtraction, scalar multiplication, and matrix multiplication. However, strict rules govern these operations:

• Matrix addition and subtraction require matrices of identical dimensions.
• For scalar multiplication, the matrix's size remains unchanged as each element is multiplied by a scalar value.
• Matrix multiplication involves multiplying rows by columns and is only feasible when matrix dimensions permit as per compatibility rules.

Consider solving matrix-related problems like solving puzzles, where each piece (or matrix) fits together in a specific way to reveal the complete picture. Properly identifying possibilities and limits within matrix algebra is key to executing these operations effectively.