Problem 29 - 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\\. \(4(A+B)=4 A+4 B\)

Step-by-Step Solution

Answer

We have verified the given equation through direct computation as follows: 1. Calculated the sum of matrices A and B: \(A+B=\left[\begin{array}{rr}4 & 3 \\\1 & 4 \\\-1 & 2\end{array}\right]\) 2. Calculated 4 times the sum: \(4(A+B)=\left[\begin{array}{rr}16 & 12 \\\4 & 16 \\\-4 & 8\end{array}\right]\) 3. Calculated 4 times A and 4 times B: \(4A=\left[\begin{array}{rr}12 & 4 \\\8 & 16 \\\-16 & 0\end{array}\right]\) and \(4B=\left[\begin{array}{rr}4 & 8 \\\-4 & 0 \\\12 & 8\end{array}\right]\) 4. Calculated the sum of 4A and 4B: \(4A+4B=\left[\begin{array}{rr}16 & 12 \\\4 & 16 \\\-4 & 8\end{array}\right]\) Since both calculated values 4(A+B) and 4A+4B are equal: \(4(A+B) = 4A + 4B\)

Step 1: Calculate A + B

Add matrices A and B by adding their corresponding elements: $$A+B = \left[\begin{array}{rr}3 & 1 \\\2 & 4 \\\\-4 & 0\end{array}\right] + \left[\begin{array}{rr}1 & 2 \\\\-1 & 0 \\\3 & 2\end{array}\right] = \left[\begin{array}{rr}3+1 & 1+2 \\\2+-1 & 4+0 \\\-4+3 & 0+2\end{array}\right]=\left[\begin{array}{rr}4 & 3 \\\1 & 4 \\\-1 & 2\end{array}\right]$$

Step 2: Calculate 4(A + B)

Multiply each element of the sum (A+B) by 4: $$4(A+B) = 4\left[\begin{array}{rr} 4 & 3 \\\1 & 4 \\\-1 & 2\end{array}\right] = \left[\begin{array}{rr} 4(4) & 4(3) \\\4(1) & 4(4) \\\4(-1) & 4(2)\end{array}\right] = \left[\begin{array}{rr} 16 & 12 \\\4 & 16 \\\-4 & 8\end{array}\right]$$

Step 3: Calculate 4A and 4B

Multiply each element of matrix A and matrix B by 4: $$4A = \left[\begin{array}{rr} 4(3) & 4(1) \\\4(2) & 4(4) \\\4(-4) & 4(0)\end{array}\right] = \left[\begin{array}{rr} 12 & 4 \\\8 & 16 \\\-16 & 0\end{array}\right]$$ $$4B = \left[\begin{array}{rr} 4(1) & 4(2) \\\4(-1) & 4(0) \\\4(3) & 4(2)\end{array}\right] = \left[\begin{array}{rr} 4 & 8 \\\-4 & 0 \\\12 & 8\end{array}\right]$$

Step 4: Calculate 4A + 4B

Add matrices 4A and 4B by adding their corresponding elements: $$4A+4B = \left[\begin{array}{rr}12 & 4 \\\8 & 16 \\\-16 & 0\end{array}\right] +\left[\begin{array}{rr}4 & 8 \\\-4 & 0 \\\12 & 8\end{array}\right]=\left[\begin{array}{rr}12+4 & 4+8 \\\8+-4 & 16+0 \\\-16+12 & 0+8\end{array}\right]=\left[\begin{array}{rr}16 & 12 \\\4 & 16 \\\-4 & 8\end{array}\right]$$

Step 5: Verify if 4(A+B) and 4A + 4B are equal

Compare the results from step 2 and step 4: $$4(A+B) = \left[\begin{array}{rr} 16 & 12 \\\4 & 16 \\\-4 & 8\end{array}\right]$$ $$4A + 4B = \left[\begin{array}{rr} 16 & 12 \\\4 & 16 \\\-4 & 8\end{array}\right]$$ Since 4(A+B) and 4A + 4B are equal, the given equation is verified: $$4(A+B) = 4A + 4B$$

Key Concepts

Direct Computation in Algebra

Direct computation in algebra involves performing operations exactly as they are presented, following the conventional order of operations. When applied to matrices, this means calculating results by individually addressing the components of the matrices involved.

For example, to verify the equation in the textbook problem, we must execute each operation step by step. First, we add matrices A and B to find matrix (A+B), then we multiply this sum by the scalar quantity '4'. This process of direct computation is crucial in linear algebra, particularly when verifying matrix equations or simplifying matrix expressions. It's the mathematical equivalent of 'showing your work', to ensure that each step follows the rules and properties of algebraic operations.

Matrix Operations

Matrix operations, such as addition and scalar multiplication, are fundamental in understanding how to manipulate and utilize matrices in various fields of study, from physics to economics.

In matrix addition, as seen in the provided exercise, two matrices of the same dimensions can be added together by summing their corresponding elements. Scalar multiplication involves multiplying every element of a matrix by a scalar value. For instance, to scalarly multiply a matrix by 4, every single element in the matrix is multiplied by 4, producing a new matrix with the same dimensions. Learning these operations is crucial for anyone aiming to understand more complex matrix manipulations.

Properties of Matrix Operations

Just like basic arithmetic, matrix operations abide by certain properties that define how these operations can be conducted. For instance, matrix addition is commutative (A + B = B + A) and associative ((A + B) + C = A + (B + C)). Scalar multiplication is distributive over matrix addition, which is the property being verified in the exercise: 4(A + B) = 4A + 4B.

This means that multiplying a scalar with a sum of matrices is the same as multiplying the scalar with each matrix individually and then adding the results. Understanding these properties is not only key to performing matrix operations correctly but also to understanding the structure and behavior of linear systems.

Verifying Matrix Equations

Verifying matrix equations through direct computation is not only a proof of mastery of matrix operations, but it also solidifies one's understanding of algebraic principles. The backbone of verifying such equations is rooted in properties of matrix operations.

For the given problem, to verify the equation 4(A+B) = 4A + 4B, we first compute 4(A+B) and then 4A + 4B separately, by following matrix addition and scalar multiplication rules. After calculating both sides of the equation, if each corresponding element is equal, then the equation holds true. This practice of verification is essential in many mathematical and engineering disciplines, as it provides a fundamental technique for ensuring the integrity of computations and derivations.