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Chapter 3: Problem 63

Why is matrix addition associative?

Short Answer

Expert verified
Matrix addition is associative because for any three matrices A, B, and C of the same size, the result of the addition is the same for both expressions (A + B) + C and A + (B + C), element by element. This is due to the associativity of real numbers in individual matrix elements, which leads to the equality \(e_{ij} = g_{ij}\) for all i and j, where \(e_{ij} = (a_{ij} + b_{ij}) + c_{ij}\) and \(g_{ij} = a_{ij} + (b_{ij} + c_{ij})\).

Step by step solution

01

Set up the given information

We have three matrices A, B and C: - \(A = [a_{ij}]_{m \times n}\) with elements \(a_{ij}\) for i = 1,...,m and j = 1,...,n - \(B = [b_{ij}]_{m \times n}\) with elements \(b_{ij}\) for i = 1,...,m and j = 1,...,n - \(C = [c_{ij}]_{m \times n}\) with elements \(c_{ij}\) for i = 1,...,m and j = 1,...,n
02

Calculate (A + B) + C

Add matrices A and B element by element to get a new matrix D: - \(D = A + B = [d_{ij}]_{m \times n}\) with elements \(d_{ij} = a_{ij} + b_{ij}\) Next, add matrix D to C element by element: - \((A + B) + C = D + C = [e_{ij}]_{m \times n}\), with elements \(e_{ij} = d_{ij} + c_{ij} = (a_{ij} + b_{ij}) + c_{ij}\)
03

Calculate A + (B + C)

Add matrices B and C element by element to get a new matrix E: - \(E = B + C = [f_{ij}]_{m \times n}\) with elements \(f_{ij} = b_{ij} + c_{ij}\) Next, add matrix A to E: - \(A + (B + C) = A + E = [g_{ij}]_{m \times n}\), with elements \(g_{ij} = a_{ij} + f_{ij} = a_{ij} + (b_{ij} + c_{ij})\)
04

Compare the results

We must show that \(e_{ij} = g_{ij}\) for all i and j. From Steps 2 and 3, we have: - \(e_{ij} = (a_{ij} + b_{ij}) + c_{ij}\) - \(g_{ij} = a_{ij} + (b_{ij} + c_{ij})\) Due to the associativity of real numbers, we can rewrite the expressions as follows: - \(e_{ij} = a_{ij} + b_{ij} + c_{ij}\) - \(g_{ij} = a_{ij} + b_{ij} + c_{ij}\) Since \(e_{ij} = g_{ij}\) for all i and j, we can conclude that (A + B) + C = A + (B + C) for any matrices A, B, and C of the same size. Therefore, matrix addition is associative.

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Most popular questions from this chapter

What would a matrix \(A\) look like if \(A_{i j}=0\) whenever \(i \neq j\) ?

Y o u ~ a n d ~ y o u r ~ f r i e n d ~ h a v e ~ c o m e ~ u p ~ with the following simple game to pass the time: at each round, you simultaneously call "heads" or "tails." If you have both called the same thing, your friend wins one point; if your calls differ, you win one point.

Evaluate the given expression. Take \(A=\left[\begin{array}{rrr}1 & -1 & 0 \\\ 0 & 2 & -1\end{array}\right], B=\left[\begin{array}{rrr}3 & 0 & -1 \\ 5 & -1 & 1\end{array}\right]\), and \(C=\left[\begin{array}{lll}x & 1 & w \\ z & r & 4\end{array}\right] .\) $$ 2 A-B $$

What does it mean when we say that \((A+B)_{i j}=A_{i j}+B_{i j}\) ?

Translate the given systems of equations into matrix form. \(x-y=4\) \(2 x-y=0\)
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