Calculate matrix?
2 Answers
Explanation:
#"two matrices are "color(blue)"conformable"# for multiplication if the number of columns in the first is equal to the number of rows in the second.
#"given order "(color(magenta)(m)xxcolor(blue)(p))xx(color(blue)(p)xxcolor(red)(n))#
#"then order of multiplication is "(color(magenta)(m)xxcolor(red)(n))#
#"A is order "(3xx2)" and B is of order "(2xx2)#
#"thus they are conformable for multiplication and order"#
#"of product is "(3xx2)#
#AB=((1/3,-2),(1,0),(0,-3))((9,1),(-1,1/2))#
#"multiply corresponding elements in rows of A with elements"#
#"in columns of B and sum them"#
#color(white)(AB)=((3+2,1/3-1),(9+0,1+0),(0+3,0-3/2))=((5,-2/3),(9,1),(3,-3/2))#
#BA-"B is of order " (2xx2) and "A is "(3xx2)#
#"thus BA is not conformable for multiplication"#
#A^2" is order "(3xx2)xx(3xx2)#
#"and is not conformable for multiplication"#
#AB-A#
#=((5,-2/3),(9,1),(3,-3/2))-((1/3,-2),(1,0),(0,-3))#
#"subtract corresponding elements in each matrix"#
#=((5-1/3,-2/3-2),(9-1,1-0),(3-0,-3/2+3))#
#=((14/3,-8/3),(8,1),(3,3/2))#
#-3A#
#=color(red)(-3)((1/3,-2),(1,0),(0,-3))#
#"multiply each element of the matrix by "-3#
#=((-1,2/3),(-3,0),(0,9))#
#"parts b and c on separate answer"#
Explanation:
#(b)#
#"given a "2xx2" matrix "A=((a,b),(c,d))#
#"then inverse matrix "A^-1=1/(ad-bc)((d,-b),(-c,a))#
#A^-1" only exists if "ad-bc!=0#
#((2,3),(4,5))#
#"has "color(blue)"determinant "ad-bc=(5xx2)-(3xx4)=-2#
#rArr"is invertible"#
#"inverse "=-1/2((5,-3),(-4,2))=((-5/2,3/2),(2,-1))#
#((4,-2),(6,-3))#
#rArrad-bc=(4xx-3)-(-2xx6)=0#
#rArr((4,-2),(6,-3))" is not invertible"#
#(c)#
#"we require to solve "((2,3),(4,5))((x),(y))=((4),(2))#
#"that is "AX=B#
#rArrA^-1AX=A^-1B#
#rArrIX=A^-1B#
#rArrX=A^-1B#
#"from "(b)toA^-1=((-5/2,3/2),(2,-1))#
#rArrX=((-5/2,3/2),(2,-1))((4),(2))#
#color(white)(rArrX)=((-10+3),(8-2))=((-7),(6))#
#rArrx=-7" and "y=6#
