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

Step-by-Step Solution

Answer

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

Step 1: Identify dimensions of matrices B, F, and E

Matrix \( B \) is a \( 2 \times 3 \) matrix. Matrix \( F \) is a \( 3 \times 3 \) identity matrix. Matrix \( E \) is a \( 3 \times 1 \) column vector.

Step 2: Determine the possibility of multiplying B and F

To multiply two matrices, the number of columns in the first matrix must be equal to the number of rows in the second matrix. Here, \( B \) has 3 columns, which matches the 3 rows of \( F \). Therefore, the multiplication \( BF \) is possible and results in a \( 2 \times 3 \) matrix.

Step 3: Perform the multiplication of B and F

Since \( F \) is the identity matrix, multiplying \( B \) with \( F \) results in \( B \) itself. Thus, \( BF = B = \begin{bmatrix} 3 & \frac{1}{2} & 5 \ 1 & -1 & 3 \end{bmatrix} \).

Step 4: Determine the possibility of multiplying F and E

Check the dimensions for the multiplication \( FE \). Matrix \( F \) has 3 columns, and \( E \) has 3 rows, so this multiplication is possible and will result in a \( 3 \times 1 \) column vector.

Step 5: Perform the multiplication of F and E

Multiplying \( F \), the identity matrix, by \( E \) will result in \( E \) being unchanged: \( FE = E = \begin{bmatrix} 1 \ 2 \ 0 \end{bmatrix} \).

Step 6: Determine if the addition BF + FE is possible

Adding two matrices is only possible if they have the same dimensions. Here, \( BF \) is a \( 2 \times 3 \) matrix, and \( FE \) is a \( 3 \times 1 \) column vector. Since these matrices do not have matching dimensions, the operation \( BF + FE \) is not defined.

Key Concepts

Matrix Dimensions

Understanding matrix dimensions is fundamental to a range of matrix operations. The dimensions of a matrix are given as "rows x columns", where rows represent the number of horizontal lines of entries, and columns are the vertical lines of entries. For example, in the given exercise:

• Matrix \( B \) is a \( 2 \times 3 \) matrix, meaning it has 2 rows and 3 columns.
• Matrix \( F \) is a \( 3 \times 3 \) matrix, an identity matrix with 3 rows and 3 columns.
• Matrix \( E \) is a column vector with dimensions \( 3 \times 1 \), having 3 rows and 1 column.

Understanding these dimensions is necessary for performing operations like matrix multiplication and addition. Always check dimensions before attempting operations, as incorrect dimensions can lead to undefined results.

Matrix Multiplication

Matrix multiplication is a bit more complex than simple arithmetic multiplication. For two matrices to be multiplied, their dimensions must align correctly. Specifically, the number of columns in the first matrix needs to match the number of rows in the second matrix.For instance, multiplying matrix \( B \) (\( 2 \times 3 \)) with matrix \( F \) (\( 3 \times 3 \)) is possible because the number of columns in \( B \) (3) matches the number of rows in \( F \) (3). The resulting product, \( BF \), will have the dimensions of the outer numbers (i.e., \( 2 \times 3 \)). Matrix multiplication is non-commutative, meaning \( AB \) is not necessarily equal to \( BA \). It is also important to know that multiplying by an identity matrix, such as matrix \( F \), will leave the original matrix unchanged.

Identity Matrix

The identity matrix is a special type of matrix that plays a crucial role in matrix operations, akin to the number 1 in multiplication. An identity matrix is a square matrix (i.e., it has the same number of rows and columns) with ones on its main diagonal and zeros elsewhere.In our exercise, matrix \( F \) is a \( 3 \times 3 \) identity matrix. When you multiply any matrix by an identity matrix that matches its dimensions, the result remains unchanged. For example:

• Multiplying \( B \) by the identity matrix \( F \) (where appropriate dimensions match) yields \( B \) itself, represented as \( BF = B \).
• Similarly, multiplying the column vector \( E \) by \( F \) results in \( E \) itself, noted as \( FE = E \).

The identity matrix is fundamental in maintaining the integrity of a matrix during transformations.

Matrix Addition

Matrix addition involves adding corresponding elements of two matrices together. However, this operation is only possible if both matrices have the exact same dimensions.In our exercise, we needed to attempt the addition of matrices \( BF \) and \( FE \). Matrix \( BF \) had dimensions \( 2 \times 3 \), while matrix \( FE \) was \( 3 \times 1 \). Since their dimensions do not match, the addition could not be performed. Simple rules for matrix addition:

• Both matrices must have the same number of rows and columns.
• Add corresponding elements together to form a new matrix.

If the matrices do not align in dimension, then matrix addition is undefined, as demonstrated in this problem.