StudentAidUnit 3: Matrices and Determinants — Exercise 3 2
10th Class Mathematics · Unit 3: Matrices and Determinants
3.2.1.From the following matrices identify unit matrices, row matrices, column matrices and null matrices: A=[5,7,8], B=[0], C=[[1,0],[0,1]], D=[[0,0],[0,0]], E=[[6],[0],[8]], F=[[7],[1],[9]]
Result
Identification
Unit: C; Row: A; Column: B,E,F; Null: D
3.2.2.Identify type of the given matrices as row, column, square and rectangular matrices: A=[[5],[3],[4]], B=[[1,6],[4,1]], C=[[3,5,8],[0,4,-2]], D=[[5,-5],[2,7]], E=[[3,2],[4,1],[5,0]], F=[5,-3,7], G=[[3,0,1],[1,3,4],[5,2,-3]], H=[[3,5],[4,4],[5,2]]
Result
Identification
A:Column (3 × 1); B:Square (2 × 2); C:Rectangular (2 × 3); D:Square (2 × 2); E:Rectangular (3 × 2); F:Row (1 × 3); G:Square (3 × 3); H:Rectangular (3 × 2)
3.2.3.Identify diagonal, scalar and unit matrices: A=[[1,0],[0,1]], B=[[5,0],[0,5]], C=[[3,0],[0,0]]
Result
Identification
Diagonal: A,B,C;quad Scalar: B;quad Unit: A
3.2.4.Find transpose of each of the following matrices: A=[[2,3],[4,5]], B=[[3],[7],[6]], C=[5,-2,4], D=[[2,5],[3,6],[4,7]]
Result
Transposes
At=begin{bmatrix}2&43&5end{bmatrix}, Bt=begin{bmatrix}3&7&6end{bmatrix}, Ct=begin{bmatrix}5-24end{bmatrix}, Dt=begin{bmatrix}2&3&45&6&7end{bmatrix}
3.2.5.Find negative of the following matrices: A=[[-3,0],[5,6]], B=[[-3,3],[-2,2]], C=[[-9,1],[1,-7]]
Result
Negatives
-A=begin{bmatrix}3&0-5&-6end{bmatrix}, -B=begin{bmatrix}3&-32&-2end{bmatrix}, -C=begin{bmatrix}9&-1-1&7end{bmatrix}
3.2.6.(i) If A=[[2,3],[4,5]] and B=[[7,6],[5,8]], verify that (A^t)^t = A. (ii) verify that (B^t)^t = B.
(i)
At=begin{bmatrix}2&43&5end{bmatrix} ⇒ (At)t=begin{bmatrix}2&34&5end{bmatrix}=A
Result
(i) Verified
(At)t=A
(ii)
Bt=begin{bmatrix}7&56&8end{bmatrix} ⇒ (Bt)t=begin{bmatrix}7&65&8end{bmatrix}=B
(ii) Verified
(Bt)t=B
3.2.7.Show that L=[[2,3,4],[3,-2,5],[4,5,0]] is a symmetric matrix.
Transpose
Lt=begin{bmatrix}2&3&43&-2&54&5&0end{bmatrix}
Result
Conclusion
Lt=L ⇒ L is symmetric