Solution: A = ቂ 7 −2 −3 10 5 8 0 1 ቃ , B = ቂ 8 −1 4 −2 ቃ , C = −6 3 0 4 9 −2 8 5 , D = ቈ 5 −3 2 0 8 6 1 −2 −6 a) A2 x 4, B2 x 2; number of columns of A (4) is not equal number of rows of B (2). Therefore, AB does not exist. b) A T = 7 −3 −2 10 5 0 8 1 , A T B = 7 −3 −2 10 5 0 8 1 ቂ 8 −1 4 −2 ቃ= ⎣ ⎢ ⎢ ⎡ 7 ∗ 8 + (−3) ∗ (4) 7 ∗ (−1) + (−3) ∗ (−2) −2 ∗ 8 + 10 ∗ 4 −2 ∗ (−1) + 10 ∗ (−2) 5∗8+0∗4 5 ∗ (−1) + 0 ∗ (−2) 8∗8+1∗4 8 ∗ (−1) + 1 ∗ (−2) ⎦ ⎥ ⎥ ⎤ = 44 −1 24 −18 40 −5 68 −10 . c) AC= ቂ 7 −2 −3 10 5 8 0 1 ቃ −6 3 0 4 9 −2 8 5 = 7 ∗ (−6) + (−2) ∗ 0 + 5 ∗ 9 + 8 ∗ 8 7 ∗ 3 + (−2) ∗ 4 + 5 ∗ (−2) + 8 ∗ 5 (−3) ∗ (−6) + 10 ∗ 0 + 0 ∗ 9 + 1 ∗ 8 (−3) ∗ (3) + 10 ∗ 4 + 0 ∗ (−2) + 1 ∗ 5 ൨= =ቂ 67 43 26 36 ቃ d) 2B +3I2 = 2 ቂ 8 −1 4 −2 ቃ+3 ቂ 1 0 0 1 ቃ= 2∗8+3∗1 2 ∗ (−1) + 0 2∗4+3∗0 2 ∗ (−2) + 3 ∗ 1 ൨ = ቂ 19 −2 8 −1 ቃ e) det ቈ 5 −3 2 0 8 6 1 −2 −6 = ቈ 5 −3 2 0 8 6 1 −2 −6 5 −3 2 0 1 −2 = 5*0*(-6)+(-3)*6*1+8*2*(-2) – 8*0*1- 5*6*(-2)–(-3)*2*(-6) = -26. f) detቂ 8 −1 4 −2 ቃ = 8 ∗ (−2) − (−1) ∗ 4 = −12. B -1 = 1/(-12) ቂ −2 1 −4 8 ቃ = 1/6 −1/12 1/3 −2/3 ൨ .