THE THEORY OF MATRICES AND DETERMINANTS 269F
Problem 5. IfA=(
23
1 − 4)
andB=(
− 57
− 34)findA×B.LetA×B=CwhereC=
(
C 11 C 12
C 21 C 22)C 11 is the sum of the products of the first row ele-
ments ofAand the first column elements ofBtaken
one at a time,
i.e. C 11 =(2×(−5))+(3×(−3))=− 19
C 12 is the sum of the products of the first row ele-
ments ofAand the second column elements ofB,
taken one at a time,
i.e. C 12 =(2×7)+(3×4)= 26
C 21 is the sum of the products of the second row
elements ofAand the first column elements ofB,
taken one at a time,
i.e. C 21 =(1×(−5))+(− 4 ×(−3))= 7
Finally,C 22 is the sum of the products of the second
row elements ofAand the second column elements
ofB, taken one at a time,
i.e. C 22 =(1×7)+((−4)×4)=− 9
Thus,A×B=
(
− 19 26
7 − 9)Problem 6. Simplify
(
340
− 26 − 3
7 − 41)×(
2
5
− 1)The sum of the products of the elements of each
row of the first matrix and the elements of the second
matrix, (called acolumn matrix), are taken one at a
time. Thus:
(
340
− 26 − 3
7 − 41
)×(
2
5
− 1)=(
(3×2) +(4×5) +(0×(−1))
(− 2 ×2)+(6×5) +(− 3 ×(−1))
(7×2) +(− 4 ×5)+(1×(−1)))=(
26
29
− 7)Problem 7. IfA=(
340
− 26 − 3
7 − 41)andB=(
2 − 5
5 − 6
− 1 − 7), findA×B.The sum of the products of the elements of each row
of the first matrix and the elements of each column of
the second matrix are taken one at a time. Thus:
(
340
− 26 − 3
7 − 41)×(
2 − 5
5 − 6
− 1 − 7)=⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝
[(3×2) [(3×(−5))
+(4×5) +(4×(−6))
+(0×(−1))] +(0×(−7))]
[(− 2 ×2) [(− 2 ×(−5))
+(6×5) +(6×(−6))
+(− 3 ×(−1))] +(− 3 ×(−7))]
[(7×2) [(7×(−5))
+(− 4 ×5) +(− 4 ×(−6))
+(1×(−1))] +(1×(−7))]⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ =(
26 − 39
29 − 5
− 7 − 18)Problem 8. Determine
(
103
212
131)×(
220
132
320)The sum of the products of the elements of each row
of the first matrix and the elements of each column of
the second matrix are taken one at a time. Thus:
(
103
212
131)×(
220
132
320)=⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝
[(1×2) [(1×2) [(1×0)
+(0×1) +(0×3) +(0×2)
+(3×3)] +(3×2)] +(3×0)]
[(2×2) [(2×2) [(2×0)
+(1×1) +(1×3) +(1×2)
+(2×3)] +(2×2)] +(2×0)]
[(1×2) [(1×2) [(1×0)
+(3×1) +(3×3) +(3×2)
+(1×3)] +(1×2)] +(1×0)]⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ =(
11 8 0
11 11 2
8136)