Diketahui A = (1 3 3 4), B = (3 -1 2 0) dan C = (-1 2 1 4)

a. [[10, 0], [28, -4]], b. [[5, 13], [3, 22]], c. [[11, 13], [18, 19]], d. [[5, 80], [10, 130]], e. [[15, -7], [14, -6]]

Pembahasan

Diberikan matriks A = [[1, 3], [3, 4]], B = [[3, -1], [2, 0]], dan C = [[-1, 2], [1, 4]].

a. AB + CB
AB = [[1*3+3*2, 1*(-1)+3*0], [3*3+4*2, 3*(-1)+4*0]] = [[3+6, -1+0], [9+8, -3+0]] = [[9, -1], [17, -3]]
CB = [[-1*3+2*2, -1*(-1)+2*0], [1*3+4*2, 1*(-1)+4*0]] = [[-3+4, 1+0], [3+8, -1+0]] = [[1, 1], [11, -1]]
AB + CB = [[9+1, -1+1], [17+11, -3+(-1)]] = [[10, 0], [28, -4]]

b. AC + B
AC = [[1*(-1)+3*1, 1*2+3*4], [3*(-1)+4*1, 3*2+4*4]] = [[-1+3, 2+12], [-3+4, 6+16]] = [[2, 14], [1, 22]]
AC + B = [[2+3, 14+(-1)], [1+2, 22+0]] = [[5, 13], [3, 22]]

c. A(C + B)
C + B = [[-1+3, 2+(-1)], [1+2, 4+0]] = [[2, 1], [3, 4]]
A(C + B) = [[1*2+3*3, 1*1+3*4], [3*2+4*3, 3*1+4*4]] = [[2+9, 1+12], [6+12, 3+16]] = [[11, 13], [18, 19]]

d. A^2C
A^2 = A * A = [[1*1+3*3, 1*3+3*4], [3*1+4*3, 3*3+4*4]] = [[1+9, 3+12], [3+12, 9+16]] = [[10, 15], [15, 25]]
A^2C = [[10*(-1)+15*1, 10*2+15*4], [15*(-1)+25*1, 15*2+25*4]] = [[-10+15, 20+60], [-15+25, 30+100]] = [[5, 80], [10, 130]]

e. B^3
B^2 = B * B = [[3*3+(-1)*2, 3*(-1)+(-1)*0], [2*3+0*2, 2*(-1)+0*0]] = [[9-2, -3+0], [6+0, -2+0]] = [[7, -3], [6, -2]]
B^3 = B^2 * B = [[7*3+(-3)*2, 7*(-1)+(-3)*0], [6*3+(-2)*2, 6*(-1)+(-2)*0]] = [[21-6, -7+0], [18-4, -6+0]] = [[15, -7], [14, -6]]