Problem 28 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Let $$A=\left[\begin{array}{rr}3 & 1 \\\2 & 4 \\\\-4 & 0\end{array}\right] \text { and } B=\left[\begin{array}{rr}1 & 2 \\\\-1 & 0 \\\3 & 2\end{array}\right]$$ Verify each equation by direct computation\\. \(2(4 A)=(2 \cdot 4) A=8 A\)

Step-by-Step Solution

Answer

The equation \(2(4A) = (2 \cdot 4)A = 8A\) is verified, as both scalar multiplications result in the same matrix: \(\left[\begin{array}{rr}24 & 8 \\ 16 & 32 \\ -32 & 0\end{array}\right]\).

Step 1: Determine the scalar multiplication of 2(4A)

First, we need to find the scalar multiplication of 4A. To do that, multiply each entry of matrix A by the scalar 4: \(4A = \left[\begin{array}{rr}12 & 4 \\ 8 & 16 \\ -16 & 0\end{array}\right]\) Now, we need to find the scalar multiplication of 2(4A). To do that, multiply each entry of matrix (4A) by the scalar 2: \(2(4A) = 2 \left[\begin{array}{rr}12 & 4 \\ 8 & 16 \\ -16 & 0\end{array}\right] = \left[\begin{array}{rr}24 & 8 \\ 16 & 32 \\ -32 & 0\end{array}\right]\)

Step 2: Determine the scalar multiplication of (2 · 4)A = 8A

Now, we need to find the scalar multiplication of 8A. To do that, multiply each entry of matrix A by the scalar 8: \(8A = \left[\begin{array}{rr}24 & 8 \\ 16 & 32 \\ -32 & 0\end{array}\right]\)

Step 3: Compare the results and verify the equation

Now that we have the results of both scalar multiplications, we can compare them to check if they are the same: 2(4A) = \(\left[\begin{array}{rr}24 & 8 \\ 16 & 32 \\ -32 & 0\end{array}\right]\) (2 · 4)A = 8A = \(\left[\begin{array}{rr}24 & 8 \\ 16 & 32 \\ -32 & 0\end{array}\right]\) Since the results of both scalar multiplications are the same, the equation \(2(4A) = (2 \cdot 4)A = 8A\) is verified.

Key Concepts

Scalar Multiplication

Scalar multiplication is a fundamental operation in matrix algebra that involves multiplying each element of a matrix by a constant, known as a scalar. This operation is quite simple yet crucial for many matrix transformations. Let's delve deeper into understanding scalar multiplication with an example.

Suppose we have a matrix \(A\) and a scalar \(k\). To perform scalar multiplication, we multiply every entry of matrix \(A\) by the scalar \(k\).

For example, if \(A = \left[\begin{array}{rr}3 & 1 \2 & 4 \-4 & 0\end{array}\right]\) and \(k = 4\), then \(4A\) is computed by multiplying each entry of \(A\) by 4.

This results in:

• First row: \(3 \times 4 = 12\) and \(1 \times 4 = 4\)
• Second row: \(2 \times 4 = 8\) and \(4 \times 4 = 16\)
• Third row: \(-4 \times 4 = -16\) and \(0 \times 4 = 0\)

Thus, \(4A = \left[\begin{array}{rr}12 & 4 \8 & 16 \-16 & 0\end{array}\right]\).

This simple procedure allows us to scale matrices, making scalar multiplication a foundational step in more complex operations like matrix multiplication.

Matrix Multiplication

Matrix multiplication is a key operation in linear algebra that combines two matrices to produce a new matrix. Unlike scalar multiplication, where each entry is scaled by a single number, matrix multiplication involves a more intricate process where rows and columns are combined.

To multiply two matrices \(A\) and \(B\), you follow these basic principles:

• Ensure that the number of columns in the first matrix \(A\) matches the number of rows in the second matrix \(B\).
• The resulting matrix has dimensions equal to the number of rows of \(A\) by the number of columns of \(B\).

For example, if matrix \(A\) is 3x2 and matrix \(B\) is 2x3, you can multiply them to get a 3x3 matrix.

The multiplication is done by computing the dot product of rows of the first matrix with columns of the second matrix. This involves:

• Taking each row element from matrix \(A\), multiplying by the corresponding column element in matrix \(B\), and summing these products.
• This sum forms an entry in the resulting matrix.

This operation is essential for many applications in engineering, physics, and computer graphics, where transformations and data processing tasks often rely on these matrix interactions.

Verification of Equations

Verification of equations in matrix operations ensures that the mathematical expressions or transformations are valid and consistent. This is especially important in solving problems involving matrices where calculations might become complex.

One common verification task is checking whether an equation involving matrices holds true after performing operations like addition, multiplication, or scalar multiplication.

Let's consider an example: verifying the equation \(2(4A) = (2 \cdot 4)A\).

To verify, you perform the following steps:

• Calculate \(4A\) by multiplying each element in matrix \(A\) by 4.
• Then, calculate \(2(4A)\) by scaling the result from the previous step by 2.
• Independently, compute \(8A = (2 \cdot 4)A\) by directly multiplying every entry of \(A\) by 8.

If both results from the above steps match, the equation is verified. This ensures that the operations performed obey the algebraic rules and help establish confidence in solving more complex matrix equations.