Problem 26 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Let $$\begin{array}{l}A=\left[\begin{array}{rrr}1 & 0 & -2 \\\1 & -3 & 2 \\\\-2 & 1 & 1 \end{array}\right] \quad B=\left[\begin{array}{rrr}3 & 1 & 0 \\\2 & 2 & 0 \\ 1 & -3 & -1\end{array}\right] \\\C=\left[\begin{array}{lll}2 & -1 & 0 \\\1 & -1 & 2 \\\3 & -2 & 1\end{array}\right]\end{array}$$ Verify the validity of the distributive law for matrix multiplication.

Step-by-Step Solution

Answer

In this problem, we were given matrices A, B, and C and asked to verify the validity of the distributive law for matrix multiplication. We computed A(B+C), AB, and AC, and observed that A(B+C) is not equal to AB+AC. Therefore, the distributive law does not hold for this particular set of matrices A, B, and C.

Step 1: Compute B+C

To verify the distributive law, we first need to compute the sum of matrices B and C. To do this, we add corresponding elements from both matrices. $$ B+C = \left[\begin{array}{rrr} 3+2 & 1+(-1) & 0+0 \\ 2+1 & 2+(-1) & 0+2 \\ 1+3 & (-3)+(-2) & (-1)+1 \end{array}\right] $$ So we get: $$ B+C = \left[\begin{array}{rrr} 5 & 0 & 0 \\ 3 & 1 & 2 \\ 4 & -5 & 0 \end{array}\right] $$

Step 2: Compute A(B+C)

Now, we need to compute the product A(B + C). To calculate matrix multiplication, we take each element of the first row of the first matrix and multiply it with corresponding elements from each successive column of the second matrix. $$ A(B+C) = A\left[\begin{array}{rrr} 5 & 0 & 0 \\ 3 & 1 & 2 \\ 4 & -5 & 0 \end{array}\right] $$ Performing the matrix multiplication, we get: $$ A(B+C) = \left[\begin{array}{rrr} 1(5)+0(3)+(-2)(4) & 1(0)+0(1)+(-2)(-5) & 1(0)+0(2)+(-2)(0) \\ 1(5)+(-3)(3)+2(4) & 1(0)+(-3)(1)+2(-5) & 1(0)+(-3)(2)+2(0) \\ -2(5)+1(3)+1(4) & (-2)(0)+1(1)+1(-5) & (-2)(0)+1(2)+1(0) \end{array}\right] $$ So the result is: $$ A(B+C) = \left[\begin{array}{rrr} -3 & 10 & 0 \\ 6 & -11 & -6 \\ 1 & -6 & 2 \end{array}\right] $$

Step 3: Compute AB

Next, we need to compute the product AB. This is calculated in the same method as before. $$ AB = A \times B = \left[\begin{array}{rrr} 1(3)+0(2)+(-2)(1) & 1(1)+0(2)+(-2)(-3) & 1(0)+0(0)+(-2)(-1) \\ 1(3)+(-3)(2)+2(1) & 1(1)+(-3)(2)+2(-3) & 1(0)+(-3)(0)+2(0) \\ -2(3)+1(2)+1(1) & (-2)(1)+1(2)+1(-3) & (-2)(0)+1(0)+1(0) \end{array}\right] $$ So the result is: $$ AB = \left[\begin{array}{rrr} -1 & 7 & 2 \\ -3 & -5 & 0 \\ 3 & -5 & 0 \end{array}\right] $$

Step 4: Compute AC

Now, we need to compute the product AC. $$ AC = A \times C = \left[\begin{array}{rrr} 1(2)+0(1)+(-2)(3) & 1(-1)+0(-1)+(-2)(-2) & 1(0)+0(2)+(-2)(1) \\ 1(2)+(-3)(1)+2(3) & 1(-1)+(-3)(-1)+2(-2) & 1(0)+(-3)(2)+2(1) \\ -2(2)+1(1)+1(3) & (-2)(-1)+1(-1)+1(-2) & (-2)(0)+1(2)+1(1) \end{array}\right] $$ So the result is: $$ AC = \left[\begin{array}{rrr} -2 & 3 & -2 \\ 9 & -6 & -6 \\ 4 & -1 & 2 \end{array}\right] $$

Step 5: Verify the validity of the distributive law

Finally, we need to compare the results of A(B+C) and AB+AC. $$ AB+AC = \left[\begin{array}{rrr} -1+(-2) & 7+3 & 2+(-2) \\ -3+9 & -5+(-6) & 0+(-6) \\ 3+4 & -5+(-1) & 0+2 \end{array}\right] $$ So the result is: $$ AB+AC = \left[\begin{array}{rrr} -3 & 10 & 0 \\ 6 & -11 & -6 \\ 7 & -6 & 2 \end{array}\right] $$ Since A(B+C) is not equal to AB+AC, the distributive law (A(B+C) = AB+AC) does not hold for this particular set of matrices A, B, and C.

Key Concepts

Matrix Multiplication

Matrix multiplication is a fundamental operation in linear algebra. When multiplying two matrices, the number of columns in the first matrix must equal the number of rows in the second matrix. Each element in the resulting matrix is calculated by taking the dot product of the corresponding row from the first matrix and the column from the second matrix.

For example, if you have a matrix \( A \) with dimensions \( m \times n \) and another matrix \( B \) with dimensions \( n \times p \), the resulting matrix \( AB \) will have the dimensions \( m \times p \). Each entry in \( AB \) is calculated as the sum of products between corresponding elements from each row of \( A \) and each column of \( B \). This operation is essential in various applications, from solving linear systems to representing transformations in graphics.

• The order of multiplication matters; \( AB \) is generally not equal to \( BA \).
• Not all pairs of matrices can be multiplied; the "inner dimensions" must match.
• Matrix multiplication allows the combination of linear transformations.

Understanding matrix multiplication is crucial for dealing with larger and more complex matrix algebra problems.

Distributive Law

The distributive law in matrix algebra describes how matrix multiplication distributes over matrix addition. Specifically, for any matrices \( A \), \( B \), and \( C \) of compatible dimensions, the distributive law states:

• \( A(B+C) = AB + AC \)
• \( (B+C)A = BA + CA \)

These relations indicate that multiplying a matrix by the sum of two matrices is the same as multiplying each matrix individually and then adding the results. This property is helpful when simplifying expressions and understanding algebraic structures involving matrices.

In practice, verifying the distributive law involves performing both matrix addition and multiplication operations, as demonstrated in the step-by-step solution. The distributive law is a fundamental property that holds for all matrix products where the operations are defined and is crucial when manipulating expressions involving multiple matrices in linear algebra contexts.

Matrix Addition

Matrix addition is a straightforward yet foundational operation in linear algebra. It involves adding two matrices of the same dimensions by adding their corresponding elements. If two matrices \( B \) and \( C \) are \( m \times n \) in size, the matrix sum \( B+C \) will also be \( m \times n \).

• Each element in the resulting matrix is the sum of corresponding elements from the two original matrices.
• This operation is commutative, meaning \( B+C = C+B \).
• It is also associative, so \( (B+C)+D = B+(C+D) \).

Matrix addition is akin to regular arithmetic addition but applied element-wise across matrices. It plays a crucial role in operations involving linear combinations of matrix rows or columns, making it a foundational concept to master in understanding more complex algebraic processes involving matrices.

Linear Algebra

Linear algebra is the branch of mathematics concerning vector spaces and the linear mappings between these spaces. It includes the study of lines, planes, and subspaces but is also concerned with properties such as computation with matrices, solving systems of linear equations, and transformations that preserve the linear structure of the vectors involved.

• Includes operations such as matrix multiplication, addition, and transposition.
• Used to solve linear systems by expressing them with matrix equations.
• Covers concepts like eigenvalues and eigenvectors, crucial for dimensionality reduction in data science.

Linear algebra is pivotal in various scientific and engineering disciplines including computer science, physics, and economics, as it provides methodologies to represent and solve problems in a linear framework. Understanding it allows the solving of complex problems in multivariate calculus and differential equations. Mastery of linear algebra provides problem-solving insights into vast areas like network theory, optimization, and even machine learning algorithms.