Problem 31 - 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. $$ G E $$

Step-by-Step Solution

Answer

The operation is not possible due to incompatible dimensions for multiplication.

Step 1: Determine Matrix Compatibility

To multiply two matrices, the number of columns in the first matrix must match the number of rows in the second matrix. Matrix \(G\) is a \(3 \times 3\) matrix and matrix \(E\) is a \(4 \times 1\) matrix. Therefore, they are not compatible for multiplication because the number of columns in \(G\) (which is 3) does not match the number of rows in \(E\) (which is 4).

Step 2: Conclude the Possibility of Multiplication

Since the matrices \(G\) and \(E\) do not satisfy the compatibility condition for matrix multiplication, the operation \(GE\) cannot be performed.

Key Concepts

Matrix Dimensions

Matrix dimensions are a fundamental aspect of working with matrices. A matrix is essentially a rectangular array of numbers, symbols, or expressions, arranged in rows and columns. Understanding matrix dimensions involves identifying the number of rows and columns that make up a matrix.
For example, if we have a matrix that looks like this:

• The matrix has 3 rows and 2 columns.
• Its dimensions are expressed as a 3x2 matrix.

Knowing the dimensions is crucial when performing operations like addition, subtraction, and multiplication. Specifically, in multiplication, the dimensions of the matrices determine whether the operation can be conducted, following specific rules related to compatibility.

Compatibility Condition

The compatibility condition is a rule that dictates whether two matrices can be multiplied. It's based on the dimensions of the matrices involved.
When you want to multiply matrix A by matrix B (noted as AB), the number of columns in matrix A must be equal to the number of rows in matrix B.
This can be remembered as the 'inner dimensions' needing to match.

• If matrix A is 2x3 and matrix B is 3x4, the multiplication can occur because "3" is the common dimension.
• The resulting matrix will have the 'outer dimensions', which in this case are 2x4.
• If the inner dimensions do not match (as with matrices G and E in our example, which are 3x3 and 4x1 respectively), the multiplication cannot be performed.

Understanding the compatibility condition ensures that you can correctly determine when matrix multiplication is feasible.

Matrix Compatibility

Matrix compatibility refers to the feasibility of performing operations between two or more matrices, primarily focused on multiplication.
For two matrices to be compatible for multiplication, the number of columns in the first matrix must match the number of rows in the second matrix.
In simpler terms, you 'match' the inside dimensions and keep the outside dimensions for the resultant matrix.

• Let's say matrix X is a 3x5 matrix and matrix Y is a 5x2 matrix.
• The multiplication XY is possible because 5 (columns of X) equals 5 (rows of Y).
• The outcome will be a 3x2 matrix.

Without compatibility, the multiplication operation is undefined, just as mismatched dimensions between matrices G and E mean they cannot be multiplied.

Matrix Operations

Matrix operations are various procedures you can perform on matrices, including addition, subtraction, multiplication, and more. Each operation requires certain conditions related to matrix dimensions and compatibility.
Matrix multiplication, arguably the most complex operation, involves several steps:

• Check dimensions to ensure a match (compatibility condition).
• Multiply corresponding elements and sum them up for each position.
• Construct the resulting matrix using these sums.

While addition and subtraction require matrices to have exactly the same dimensions, multiplication requires the compatibility of dimensions. Understanding these operations allows for more sophisticated matrix manipulations in math and other fields such as physics and computer science.