   Chapter 1.6, Problem 23E

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# Prove that the set S = { [ a − b b     a ] | a ,     b     ∈     ℝ } is closed with respect to matrix addition and multiplication in M 2 ( ℝ ) .

To determine

To prove: The set S={[abba]|a,b} is closed with respect to matrix addition and multiplication in M2().

Explanation

Given information:

The set S={[abba]|a,b}.

Formula used:

(1) Addition in Mm×n() is defined by

[aij]m×n+[bij]m×n=[cij]m×n, where cij=aij+bij.

(2) Definition: Matrix multiplication

The product of m×n matrix A over and n×p matrix B over is m×p matrix C=AB, where the element cij in row i and column j of AB is found by using the elements in row i of A, and the elements in column j of B in the following manner:

columnjofBcolumnjofCrowiofA[ai1ai2ai3ain][b1jb2jb3jbnj]=[cij]rowiofC

Where cij=ai1b1j+ai2b2j++ainbnj

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