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Verify that the following matrices and scalar satisfy the stated properties of Theorem 1.4.1. $$\begin{array}{c} A=\left[\begin{array}{rr} 3 & -1 \\ 2 & 4\end{array}\right], \quad B=\left[\begin{array}{rr} 0 & 2 \\ 1 & -4\end{array}\right] \\ C=\left[\begin{array}{rr}4 & 1 \\-3 & -2\end{array}\right], \quad a=4, \quad b=-7 \end{array}$$ (a) $a(B C)=(a B) C=B(a C)$ (b) $A(B-C)=A B-A C$ (c) $(B+C) A=B A+C A$ (d) $a(b C)=(a b) C$

Name: Problem 2, Chapter 1: Elementary Linear Algebra: Applications Version
Uploaded: 2020-10-18T20:05:44-08:00
Duration: 11 min 30 s
Channel: Elham Kordzadeh
Description: Verify that the following matrices and scalar satisfy the stated properties of Theorem 1.4.1. A=[ 3 -1 2 4 ], B=[ 0 2 1 -4 ] C=[ 4 1 -3 -2 ], a=4, b=-7 (a) a(B C)=(a B) C=B(a C) (b) A(B-C)=A B-A C (c) (B+C) A=B A+C A (d) a(b C)=(a b) C

   Verify that the following matrices and scalar satisfy the stated properties of Theorem 1.4.1.
$$\begin{array}{c}
A=\left[\begin{array}{rr}
3 & -1 \\
2 & 4\end{array}\right], \quad B=\left[\begin{array}{rr}
0 & 2 \\
1 & -4\end{array}\right] \\
C=\left[\begin{array}{rr}4 & 1 \\-3 & -2\end{array}\right], \quad a=4, \quad b=-7
\end{array}$$
(a) $a(B C)=(a B) C=B(a C)$
(b) $A(B-C)=A B-A C$
(c) $(B+C) A=B A+C A$
(d) $a(b C)=(a b) C$

Elementary Linear Algebra: Applications Version

Howard Anton, Chris… 11th Edition

Chapter 1, Problem 2

Instant Answer

Solved by Expert Elham Kordzadeh

10/18/2020

Step 1

(a) For $a(BC)=(aB)C=B(aC)$, we first calculate $BC$, $aB$, and $aC$: $$BC=\left[\begin{array}{rr} 0 & 2 \\ 1 & -4\end{array}\right]\left[\begin{array}{rr}4 & 1 \\-3 & -2\end{array}\right]=\left[\begin{array}{rr}-6 & -4 \\16 & Show more…

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Verify that the following matrices and scalar satisfy the stated properties of Theorem 1.4.1. $$\begin{array}{c} A=\left[\begin{array}{rr} 3 & -1 \\ 2 & 4\end{array}\right], \quad B=\left[\begin{array}{rr} 0 & 2 \\ 1 & -4\end{array}\right] \\ C=\left[\begin{array}{rr}4 & 1 \\-3 & -2\end{array}\right], \quad a=4, \quad b=-7 \end{array}$$ (a) $a(B C)=(a B) C=B(a C)$ (b) $A(B-C)=A B-A C$ (c) $(B+C) A=B A+C A$ (d) $a(b C)=(a b) C$

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Key Concepts

Associative Property of Scalar Multiplication

This property states that when multiplying a matrix by a product of scalars, the order in which the scalars are multiplied does not affect the outcome. That is, for any scalars a and b and any matrix C, the equality a(bC) = (ab)C holds. This means that scalar multiplication is associative, allowing us to group scalars together or distribute them as needed without changing the result.

Distributive Property of Matrix Multiplication Over Matrix Addition

Matrix multiplication distributes over matrix addition from both the left and the right. This means that for matrices A, B, and C, the equations A(B - C) = AB - AC and (B + C)A = BA + CA hold. This property ensures that when a matrix multiplies the sum or difference of two other matrices, it results in the sum or difference of the individual products, reflecting a fundamental compatibility of matrix addition and multiplication.

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