Example 1:
Find the product of the following pairs of matrices.
(a) (1 5 2)(243)(b) (28−31)(104−2)(c) (−35)(2 6)(d) (04−13)(7−2)(e) (7 −4)(−20−13)
Solution:
(a) (1 5 2)(243)←Matrices analysis1×3 and 3×1 ↓ ↓=1×1 matrix=(1×2 ⊕ 5×4 ⊕ 2×3)=(2+20+6)=(28)
(b)
(28−31)(104−2)←Matrices analysis2×2 and 2×2 ↓↓=2×2 matrix=(2×1+8×4 2×0+8×−2−3×1+1×4 −3×0+1×−2)=(34−161−2)
(c)
(−35)(2 6)←Matrices analysis2×1 and 1×2 ↓ ↓=2×2 matrix=(−3×2 −3×65×2 5×6)=(−6−181030)
(d)
(04−13)(7−2)←Matrices analysis2×2 and 2×1 ↓ ↓=2×1 matrix=(0×7+4×−2−1×7+3×−2)=(−8−13)
(e)
(7 −4)(−20−13)←Matrices analysis1×2 and 2×2 ↓↓=1×2 matrix=(7×−2+(−4×−1)7×0+(−4×3))=(−14+4 0−12)=(−10−12)
Find the product of the following pairs of matrices.
(a) (1 5 2)(243)(b) (28−31)(104−2)(c) (−35)(2 6)(d) (04−13)(7−2)(e) (7 −4)(−20−13)
Solution:
(a) (1 5 2)(243)←Matrices analysis1×3 and 3×1 ↓ ↓=1×1 matrix=(1×2 ⊕ 5×4 ⊕ 2×3)=(2+20+6)=(28)
(b)
(28−31)(104−2)←Matrices analysis2×2 and 2×2 ↓↓=2×2 matrix=(2×1+8×4 2×0+8×−2−3×1+1×4 −3×0+1×−2)=(34−161−2)
(c)
(−35)(2 6)←Matrices analysis2×1 and 1×2 ↓ ↓=2×2 matrix=(−3×2 −3×65×2 5×6)=(−6−181030)
(d)
(04−13)(7−2)←Matrices analysis2×2 and 2×1 ↓ ↓=2×1 matrix=(0×7+4×−2−1×7+3×−2)=(−8−13)
(e)
(7 −4)(−20−13)←Matrices analysis1×2 and 2×2 ↓↓=1×2 matrix=(7×−2+(−4×−1)7×0+(−4×3))=(−14+4 0−12)=(−10−12)
Example 2:
Find the values of m and n in each of the following matrix equations.
(a)(3m)(1 n)=(312−2−8)(b)(m2−31)(2n)=(124+2n)(c)(m−3−11)(−124n)=(−14−1153)
(a)(3m)(1 n)=(312−2−8)(33nmmn)=(312−2−8)m=−2, 3n=12n=4
(b)(m2−31)(2n)=(124+2n)(2m+2n−6+n)=(124+2n)−6+n=4+2nn=−102m+2n=122m+2(−10)=122m−20=122m=32m=16
(c)(m−3−11)(−124n)=(−14−1153)(−m+(−12)2m+(−3n)1+4−2+n)=(−14−1153)−m−12=−14−m=−2m=2−2+n=3n=5