Problem 27 - 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. $$ B F $$

Step-by-Step Solution

Answer

The result of \(BF\) is \(\left[ \begin{array}{ccc} 3 & \frac{1}{2} & 5 \\ 1 & -1 & 3 \end{array} \right]\).

Step 1: Check Matrix Dimensions

First, we need to verify if the multiplication operation \(BF\) is possible by checking the dimensions. Matrix \(B\) is a \(2\times 3\) matrix and matrix \(F\) is a \(3\times 3\) matrix. The inner dimensions (columns of \(B\) and rows of \(F\), both \(3\)) match, allowing the multiplication to proceed. The result will be a \(2\times 3\) matrix.

Step 2: Execute Matrix Multiplication

Now perform the matrix multiplication \(BF\) by multiplying each row of matrix \(B\) with each column of matrix \(F\) and summing the products. \[BF = \left[\begin{array}{rrr}{3} & {\frac{1}{2}} & {5} \ {1} & {-1} &{3}\end{array}\right] \left[\begin{array}{lll}{1} & {0} & {0} \ {0} & {1} & {0} \ {0} & {0} & {1}\end{array}\right]\]**First Row of Resultant Matrix:**- 1st element: \(3 \times 1 + \frac{1}{2} \times 0 + 5 \times 0 = 3\)- 2nd element: \(3 \times 0 + \frac{1}{2} \times 1 + 5 \times 0 = \frac{1}{2}\)- 3rd element: \(3 \times 0 + \frac{1}{2} \times 0 + 5 \times 1 = 5\)**Second Row of Resultant Matrix:**- 1st element: \(1 \times 1 + (-1) \times 0 + 3 \times 0 = 1\)- 2nd element: \(1 \times 0 + (-1) \times 1 + 3 \times 0 = -1\)- 3rd element: \(1 \times 0 + (-1) \times 0 + 3 \times 1 = 3\)

Step 3: Write Resulting Matrix

Compile the results into the resulting matrix:\[BF = \left[ \begin{array}{ccc}3 & \frac{1}{2} & 5 \1 & -1 & 3\end{array} \right]\]

Key Concepts

Matrix Dimensions

Understanding matrix dimensions is crucial for matrix operations like multiplication. A matrix is essentially an array of numbers arranged in rows and columns. When you describe the size of a matrix, you refer to its dimensions. For example, a matrix with dimensions \(m \times n\) has \(m\) rows and \(n\) columns.

For matrix multiplication to be possible, the number of columns in the first matrix must match the number of rows in the second matrix. This is why the dimensions are pivotal. In our example, matrix \(B\) is a \(2 \times 3\) matrix (2 rows and 3 columns) while matrix \(F\) is a \(3 \times 3\) matrix (3 rows and 3 columns). The `inner dimensions`, which are both 3, align perfectly, allowing us to carry out the multiplication.

Remember, the resulting matrix from multiplication will have dimensions formed by the `outer dimensions`: in our case, the result is a \(2 \times 3\) matrix. Always verify matrix dimensions before attempting any operations! This saves time and avoids errors.

Algebraic Operations

Algebraic operations with matrices can initially seem daunting, but they follow certain rules similar to basic algebra. The primary operations include addition, subtraction, and multiplication.

### Addition and Subtraction- Only matrices of the same dimensions can be added or subtracted.- These operations are performed by adding or subtracting corresponding elements of each matrix.

### MultiplicationMatrix multiplication isn't as straightforward as addition or subtraction. Instead of multiplying corresponding elements, you perform what's called the dot product of rows and columns:- For each element in the resulting matrix, multiply elements from rows of the first matrix by elements of columns of the second matrix.- Then, sum the products.

In the example from the exercise, you apply matrix multiplication rules: 1. Multiply each element of the row from matrix \(B\) by the corresponding element in the column of matrix \(F\).2. Add up all these products to fill the element in the resulting matrix.

These steps give us precise control over matrix manipulation and form the core of matrix algebra operations.

Matrix Algebra

Matrix algebra extends basic operations into a powerful mathematical language used extensively in various scientific domains. It's pivotal for solving systems of linear equations, transforming spaces in graphics, and much more.

### Important Rules- Associative Property: \((AB)C = A(BC)\)- Distributive Property: \(A(B + C) = AB + AC\)- In general, matrix multiplication is not commutative: \(AB eq BA\) in the majority of cases. This is unlike regular arithmetic and must be kept in mind.

### Identity and Inverse Matrices- **Identity Matrix:** Acts as the '1' in matrix algebra. Multiplying any matrix with an identity matrix leaves the matrix unchanged.- **Inverse Matrix:** Equivalent to division in scalar math. Not all matrices have inverses, and finding an inverse is part of advanced matrix algebra.

The step-by-step solution demonstrated multiplication involving the identity matrix (matrix \(F\) in the exercise), showcasing its property where the product \(BF\) yields matrix \(B\). Understanding these core concepts is making matrix algebra a powerful tool for computing and beyond.