Problem 27 - 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. $$B F$$

Step-by-Step Solution

Answer

The result of multiplying \( B \) and \( F \) is matrix \( B \) itself: \( \left[ \begin{array}{ccc} 3 & \frac{1}{2} & 5 \\ 1 & -1 & 3 \end{array} \right] \).

Step 1: Determine the Dimensions of Matrices B and F

Matrix \( B \) is a 2x3 matrix, and matrix \( F \) is a 3x3 matrix. To multiply two matrices, the number of columns in the first matrix must match the number of rows in the second matrix.

Step 2: Check Compatibility for Multiplication

The number of columns in matrix \( B \) is 3, and the number of rows in matrix \( F \) is also 3. Therefore, the matrices \( B \) and \( F \) are compatible for multiplication.

Step 3: Calculate the Product BF

Perform the matrix multiplication: \[ BF = \left[ \begin{array}{ccc} 3 & \frac{1}{2} & 5 \ 1 & -1 & 3 \end{array} \right] \left[ \begin{array}{ccc} 1 & 0 & 0 \ 0 & 1 & 0 \ 0 & 0 & 1 \end{array} \right] \] Since \( F \) is the identity matrix, the result \( BF \) is equal to \( B \): \[ BF = \left[ \begin{array}{ccc} 3 & \frac{1}{2} & 5 \ 1 & -1 & 3 \end{array} \right] \]

Step 4: Interpret the Result

Since matrix \( F \) is the identity matrix of appropriate dimensions, multiplying matrix \( B \) by \( F \) yields the original matrix \( B \). This confirms the property of identity matrices in matrix multiplication.

Key Concepts

Identity Matrix

An identity matrix is a special kind of matrix often denoted by the letter \( I \). The defining characteristic of an identity matrix is that it has 1s on its main diagonal (from the top left to the bottom right) and 0s elsewhere. Whenever you multiply any matrix by an identity matrix of compatible dimensions, the result is the original matrix

• For an identity matrix of size \( n \times n \), \( I_n \), you have 1s on the diagonal and 0s in every other position.
• This matrix acts as a multiplicative identity for matrices, much like the number 1 does for numbers.

In our specific problem, matrix \( F \) is the identity matrix of size \( 3\times 3 \). When matrix \( B \) (which is a \( 2 \times 3 \) matrix) is multiplied by \( F \), it retains its original form because the identity matrix causes no alteration during multiplication.This property is highly useful in many algebraic manipulations involving matrices, as it allows matrices to keep their properties intact when needed.

Matrix Dimensions

Matrix dimensions are fundamental in understanding how matrix operations work, especially multiplication. The dimensions of a matrix are given as "rows" by "columns", such as \( 2 \times 3 \) which denotes a matrix with 2 rows and 3 columns.

• The dimensions are crucial when determining if two matrices can be multiplied together.
• For matrix multiplication to be possible, the number of columns in the first matrix needs to be equal to the number of rows in the second matrix.

In our exercise, matrix \( B \) has dimensions \( 2 \times 3 \) and matrix \( F \) is \( 3 \times 3 \). To multiply these, the columns of \( B \) (3) match the rows of \( F \) (3), making the multiplication feasible. As a result, the product will have the dimensions of the number of rows of the first matrix and the number of columns of the second matrix, yielding a \( 2 \times 3 \) matrix in this case.

Matrix Properties

Understanding a few basic properties of matrices helps in performing operations such as addition, subtraction, and especially multiplication.

• Commutative Property: Unlike real numbers, matrix multiplication is not commutative, meaning \( AB eq BA \) in general.
• Associative Property: Matrix multiplication is associative, i.e., \( (AB)C = A(BC) \).
• Distributive Property: It is also distributive over addition, meaning \( A(B + C) = AB + AC \).

In our specific problem, the identity matrix property ensures that when we multiply any matrix \( B \) with the identity matrix \( F \), we get \( B \) itself. This property underlines the conceptual framework behind how matrices can be manipulated while preserving their intrinsic characteristics.These properties make matrices a powerful tool in various fields, from computer graphics to statistical analysis, where transformations and manipulations need to be carried out efficiently and accurately.