Question 102575
Let {{{A=(matrix(2,2,2,-1,4,3))}}} and {{{B=(matrix(1,2,3,1))}}}


<a href="#a">Part a</a>
<a href="#b">Part b</a>
<a href="#c">Part c</a>



<a name="a"></a>

a)


{{{4BA}}} Start with the given product


{{{4(matrix(1,2,3,1))(matrix(2,2,2,-1,4,3))}}} Plug in matrices A and B



{{{(matrix(1,2,4*3,4*1))(matrix(2,2,2,-1,4,3))}}} Multiply the scalar number 4 by every entry in matrix A




{{{(matrix(1,2,12,4))(matrix(2,2,2,-1,4,3))}}} Multiply



Now multiply matrices B and A. In order for this to be defined, the number of columns in B needs to be the same as the number of rows in A. Since this is true, BA is defined.




Now if you have two general matrices 

{{{(matrix(1,2,a,b))}}}{{{(matrix(2,2,c,d,e,f))}}}

We can multiply them like this 

{{{(matrix(1,2,a*c+b*e,a*d+b*f))}}} which will result in a 1x2 matrix



So for the matrices 


{{{(matrix(1,2,12,4))}}}{{{(matrix(2,2,2,-1,4,3))}}}


They can be multiplied like this: 


{{{(matrix(1,2,12*2+4*4,12*-1+4*3))}}} 


Multiply


{{{(matrix(1,2,24+16,-12+12))}}} 


which adds to 


{{{(matrix(1,2,40,0))}}}

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<a name="b"></a>

b)


{{{B^2}}} Start with the given expression


{{{B*B}}} Expand


{{{(matrix(1,2,3,1))(matrix(1,2,3,1))}}} Expand


Since the columns of B do not equal the rows of B, {{{B^2}}} is not defined. So in order to perform powers of matrices, the matrix needs to be square



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c)


<a name="c"></a>


{{{A+2B}}} Start with the given expression



{{{(matrix(2,2,2,-1,4,3))+2(matrix(1,2,3,1))}}} Plug in the given matrices



Since B does not have a second row, {{{A+2B}}} is not defined (remember, when you add two matrices, they need to be the same size)