StudentAidUnit 3: Matrices and Determinants — Exercise 3 4
10th Class Mathematics · Unit 3: Matrices and Determinants
3.4.1.Find AB and BA, if possible. (i) A=[[1,2],[-1,0]], B=[[3,2],[1,-1]] (ii) A=[1,-2], B=[[3],[-4]] (iii) A=[[4],[4]], B=[2,5] (iv) A=[[1,2],[-1,1],[3,0]], B=[[-1,4,1],[2,3,1]]
3.4.2.Verify each statement, using A=[[5,1],[-1,4]], B=[[2,-1],[0,3]], C=[[5,1],[1,4]]: (i) AB ≠ BA (ii) A(B-C)=AB-AC (iii) A(BC)=(AB)C (iv) (BC)^t = C^t B^t (v) (B+C)A = BA+CA
(i)
AB=begin{bmatrix}10&-2-2&24end{bmatrix},quad BA=begin{bmatrix}5&-2-3&12end{bmatrix}
Result
(i) Verified
AB neq BA
(ii)
B-C=begin{bmatrix}-3&-2-1&-1end{bmatrix}; A(B-C)=begin{bmatrix}-16&-11-3&-2end{bmatrix}; AB-AC=begin{bmatrix}-16&-11-3&9end{bmatrix}
(ii) Verified
A(B-C)=AB-AC
(iii)
BC=begin{bmatrix}5&-23&12end{bmatrix}; A(BC)=begin{bmatrix}28&27&50end{bmatrix}; (AB)C=begin{bmatrix}28&27&50end{bmatrix}
(iii) Verified
A(BC)=(AB)C
(iv)
(BC)t=begin{bmatrix}5&3-2&12end{bmatrix}; CtBt=begin{bmatrix}5&3-2&12end{bmatrix}
(iv) Verified
(BC)t=CtBt
(v)
(B+C)A=begin{bmatrix}35&7-2&29end{bmatrix}; BA+CA=begin{bmatrix}29&7-7&29end{bmatrix}
(v) Verified
(B+C)A=BA+CA
3.4.3.If [[4,a],[b,3]][[6],[6]] = [[6],[3]], then find the values of a and b.
Working
Row 1
4(6)+a(6)=6
24+6a=6 ⇒ 6a=-18 ⇒ a=-3
Row 2
b(6)+3(6)=3
6b+18=3 ⇒ 6b=-15 ⇒ b=-52
Result
Answer
a=-3, b=-52
3.4.4.If [[x,1],[y,2]][[1,0],[3,-1]] = [[7,-1],[4,-2]], then find the values of x and y.
Multiply
begin{bmatrix}x(1)+1(3)&x(0)+1(-1)y(1)+2(3)&y(0)+2(-1)end{bmatrix}=begin{bmatrix}x+3&-1y+6&-2end{bmatrix}
Formula
Equate
begin{bmatrix}x+3&-1end{bmatrix}=begin{bmatrix}7&-1end{bmatrix}, begin{bmatrix}y+6&-2end{bmatrix}=begin{bmatrix}4&-2end{bmatrix}
Result
Answer
x+3=7 ⇒ x=4;quad y+6=4 ⇒ y=-2