Problem 37 - 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. $$ B F+F E $$

Step-by-Step Solution

Answer

The operation \(BF + FE\) cannot be performed due to incompatible dimensions.

Step 1: Check Matrix Dimensions for Addition

Matrix addition requires both matrices to have the same dimensions. Matrix \(BF\) is the result of multiplying a \(2 \times 3\) matrix \(B\) with a \(3 \times 3\) identity matrix \(F\). This will result in another \(2 \times 3\) matrix. Matrix \(FE\) involves multiplying a \(3 \times 3\) matrix \(F\) with a \(4 \times 1\) matrix \(E\), which is not possible because the number of columns in \(F\) must equal the number of rows in \(E\) for multiplication to happen. Thus, \(FE\) cannot be computed.

Step 2: Compute \(BF\)

Matrix \(F\) is an identity matrix, so multiplying any matrix by an identity matrix of appropriate size leaves the original matrix unchanged. Thus, \(BF = B \). Therefore, \(BF = \left[ \begin{array}{ccc} 3 & \frac{1}{2} & 5 \ 1 & -1 & 3 \end{array} \right]\).

Step 3: Attempt to Compute \(FE\)

Matrix \(FE\) cannot be computed as indicated earlier. Specifically, the number of columns in \(F\) (3) does not match the number of rows in \(E\) (4), preventing any matrix multiplication from being performed.

Step 4: Conclude that \(BF + FE\) Cannot Be Computed

Since \(FE\) cannot be calculated due to mismatched dimensions, the operation \(BF + FE\) also cannot be executed. Addition requires two matrices of the same dimension, but here we conclude that \(FE\) does not exist, so addition is impossible.

Key Concepts

Matrix Multiplication

Matrix multiplication involves the process of taking two matrices and performing a specific dot product on their rows and columns. To multiply two matrices, say Matrix A and Matrix B, the number of columns in the first matrix (Matrix A) must match the number of rows in the second matrix (Matrix B). If these conditions are met, each element of the resulting matrix is calculated by taking the sum of products of the corresponding entries of the row of the first matrix and the column of the second matrix.
This operation results in a new matrix, where the dimensions are determined by the rows of the first matrix and the columns of the second matrix. For example:

• If an A matrix is of dimension 2x3 and a B matrix is of dimension 3x2, the resulting C matrix will have dimensions 2x2.
• A special case occurs with identity matrices, which have 1s on their diagonal and 0s elsewhere. When you multiply any matrix by an identity matrix of matching size, you'll end up with the original matrix itself, as demonstrated with matrix F in this exercise.
  This is because the identity matrix doesn't alter the original data. It essentially "preserves" any matrix it multiplies, similar to the role of the number 1 in basic multiplication.

Matrix Addition

Matrix addition is a simple operation that lets you combine two matrices by adding their corresponding elements. However, it's essential that the matrices involved in the addition have the same dimensions. This means that both matrices must have the same number of rows and the same number of columns; otherwise, addition is not possible.
For instance, if Matrix P is a 2x3 matrix, and you wish to add it to another matrix Q, Q must also be a 2x3 matrix. Each element in the resulting matrix is calculated by adding the corresponding elements from P and Q.

• If P is \(\begin{array}{cc} 1 & 2 \ 3 & 4 \end{array}\) and Q is \(\begin{array}{cc} 5 & 6 \ 7 & 8 \end{array}\), then P + Q will be \(\begin{array}{cc} 6 & 8 \ 10 & 12 \end{array}\).

If two matrices do not have the same dimensions, as we encountered with BF and FE in our exercise, addition cannot be performed. The need for equal dimensions ensures that every element has a corresponding pair across each matrix, streamlining the addition process.

Matrix Dimensions

Matrix dimensions tell us the size of a matrix and are described by the number of rows and columns it has. The format for expressing matrix dimensions is rows x columns. Understanding dimensions is crucial for conducting matrix operations such as addition, multiplication, and determining whether these operations are feasible with the given matrices.
Knowing the dimensions helps prepare for operations:

• In matrix multiplication, the number of columns in the first matrix must align with the number of rows in the second one. If Matrix A's dimensions are 3x2, it can only be multiplied by a matrix whose dimensions are 2xN (where N can be any integer).
• For matrix addition, both matrices must share identical dimensions. Thus, a 2x3 matrix can only be added to another 2x3 matrix.

In the exercise provided, understanding these dimensions is crucial as it determines what operations are possible. For example, Matrix B has dimensions of 2x3, meaning it cannot be added to matrix FE, which does not exist due to incompatible dimensions with matrix F. This clearly indicates the relationship between dimensions and operations possible.