Problem 8 - Precalculus

Question

In Exercises \(8-21,\) use the matrices \(A=\left[\begin{array}{ll}1 & 2 \\ 3 & 4\end{array}\right] \quad B=\left[\begin{array}{rr}0 & -3 \\ -5 & 2\end{array}\right] \quad C=\left[\begin{array}{rrr}10 & -\frac{11}{2} & 0 \\\ \frac{3}{5} & 5 & 9\end{array}\right]\) \(D=\left[\begin{array}{rr}7 & -13 \\\ -\frac{4}{3} & 0 \\ 6 & 8\end{array}\right] \quad E=\left[\begin{array}{rrr}1 & 2 & 3 \\ 0 & 4 & -9 \\ 0 & 0 & -5\end{array}\right]\) to compute the following or state that the indicated operation is undefined. $$ 7 B-4 A $$

Step-by-Step Solution

Answer

The result of \( 7B - 4A \) is \( \begin{bmatrix} -4 & -29 \\ -47 & -2 \end{bmatrix} \).

Step 1: Understand the Matrices

We are given matrices \( A \) and \( B \) and asked to find \( 7B - 4A \). Let's write down these matrices:\[A = \begin{bmatrix} 1 & 2 \ 3 & 4 \end{bmatrix}, \quad B = \begin{bmatrix} 0 & -3 \ -5 & 2 \end{bmatrix}\].Notice both matrices \( A \) and \( B \) are 2x2 matrices, which means they are compatible for addition and subtraction.

Step 2: Scalar Multiplication

We perform the scalar multiplication of each matrix with the given scalars. For \( 7B \), multiply each element of matrix \( B \) by 7:\[7B = 7 \times \begin{bmatrix} 0 & -3 \ -5 & 2 \end{bmatrix} = \begin{bmatrix} 0 \times 7 & -3 \times 7 \ -5 \times 7 & 2 \times 7 \end{bmatrix} = \begin{bmatrix} 0 & -21 \ -35 & 14 \end{bmatrix}\].Similarly, for \( 4A \), multiply each element of matrix \( A \) by 4:\[4A = 4 \times \begin{bmatrix} 1 & 2 \ 3 & 4 \end{bmatrix} = \begin{bmatrix} 1 \times 4 & 2 \times 4 \ 3 \times 4 & 4 \times 4 \end{bmatrix} = \begin{bmatrix} 4 & 8 \ 12 & 16 \end{bmatrix}\].

Step 3: Subtract the Matrices

Now, subtract \( 4A \) from \( 7B \) by subtracting corresponding elements:\[7B - 4A = \begin{bmatrix} 0 & -21 \ -35 & 14 \end{bmatrix} - \begin{bmatrix} 4 & 8 \ 12 & 16 \end{bmatrix} = \begin{bmatrix} 0 - 4 & -21 - 8 \ -35 - 12 & 14 - 16 \end{bmatrix} = \begin{bmatrix} -4 & -29 \ -47 & -2 \end{bmatrix}\].

Step 4: Conclusion

The result of the operation \( 7B - 4A \) is the matrix:\[\begin{bmatrix} -4 & -29 \ -47 & -2 \end{bmatrix}\].

Key Concepts

Scalar Multiplication

Scalar multiplication involves multiplying every element of a matrix by a constant number, known as a "scalar." This is an essential operation in matrix arithmetic and is straightforward to perform. If you have a matrix and a scalar, every element in the matrix is multiplied by that scalar.

For instance, in the exercise, we have two matrices, A and B. To find 7B, each element of matrix B is multiplied by 7. Likewise, each element of matrix A is multiplied by 4 to compute 4A. Here is how it looks step by step:

• Matrix B: \[ B = \begin{bmatrix} 0 & -3 \ -5 & 2 \end{bmatrix} \]
• 7 times Matrix B: Multiply each element by 7: \[ 7B = 7 \times \begin{bmatrix} 0 & -3 \ -5 & 2 \end{bmatrix} = \begin{bmatrix} 0 & -21 \ -35 & 14 \end{bmatrix} \]
• Matrix A: \[ A = \begin{bmatrix} 1 & 2 \ 3 & 4 \end{bmatrix} \]
• 4 times Matrix A: Multiply each element by 4: \[ 4A = 4 \times \begin{bmatrix} 1 & 2 \ 3 & 4 \end{bmatrix} = \begin{bmatrix} 4 & 8 \ 12 & 16 \end{bmatrix} \]

Using scalar multiplication, we prepare matrices for further calculations or operations such as matrix addition or subtraction.

Matrix Subtraction

Matrix subtraction allows us to subtract one matrix from another, but certain conditions must be met. Only matrices of the same dimensions can undergo subtraction. So, in our exercise, matrices A and B are both of dimension 2x2, making them compatible for subtraction. Here’s how matrix subtraction works:

• Given: \[ 7B = \begin{bmatrix} 0 & -21 \ -35 & 14 \end{bmatrix} \]
• And: \[ 4A = \begin{bmatrix} 4 & 8 \ 12 & 16 \end{bmatrix} \]
• Subtracting 4A from 7B: \[ 7B - 4A = \begin{bmatrix} 0 & -21 \ -35 & 14 \end{bmatrix} - \begin{bmatrix} 4 & 8 \ 12 & 16 \end{bmatrix} = \begin{bmatrix} 0-4 & -21-8 \ -35-12 & 14-16 \end{bmatrix} = \begin{bmatrix} -4 & -29 \ -47 & -2 \end{bmatrix} \]

In matrix subtraction, you subtract elements located at the same position from each other in the matrices involved. This means the element from row 1 column 1 of one matrix is subtracted from the element in row 1 column 1 of the other matrix and so on. Matrix subtraction is similar to addition but with subtraction replacing addition at every step.

2x2 Matrices

A 2x2 matrix fits neatly into the world of mathematics with two rows and two columns. These matrices are straightforward yet powerful in computations and are a great starting point for understanding matrix arithmetic. Here's what makes them unique:

• Structure: A 2x2 matrix is expressed in the form: \[ \begin{bmatrix} a & b \ c & d \end{bmatrix} \]
  
  where \(a, b, c,\) and \(d\) are elements of the matrix.
• Basic Operations: You can perform operations like addition, subtraction, and scalar multiplication easily since the computations involve only a small number of elements.
  For example, when subtracting one 2x2 matrix from another, simply subtract corresponding elements.
• Utility: Used in many areas of physics, engineering, and computer science; they help in solving equations, transformations, and analyzing systems.

The simplicity of 2x2 matrices makes them ideal for beginners when learning basic matrix operations and concepts such as determinance and inverse calculations, which are crucial in solving more advanced problems.