   Chapter 1.6, Problem 19E

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# Let a and b be real numbers and A and B elements of M m     ×     n ( ℝ ) . Prove the following properties of scalar multiplication. a ( b A ) = ( a b ) A . a ( A + B ) = a A + a B . ( a + b ) A = a A + b A

(a)

To determine

To prove: Let a and b be real numbers and A is element of Mm×n(), then a(bA)=(ab)A.

Explanation

Given information:

a and b be real numbers and A is element of Mm×n().

Formula used:

The product of a real number c and a matrix A=[aij] in Mm×n() is defined by c[aij]=[caij].

Proof:

Let a and b be real numbers and A is an element of Mm×n(), therefore, by definition of scalar multiplication of matrix

(b)

To determine

To prove: Let a be real number and A and B are elements of Mm×n(), then a(A+B)=aA+aB.

(c)

To determine

To prove: Let a and b be real numbers and A is element of Mm×n(), then (a+b)A=aA+bA.

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