Matrices Important Questions | Exercise 1 | Bsc 2nd Semester | Engineering Mathematics | Lecture -3
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00:00 - Lecture Intro
00:39 Q1.(i) Show that the matrix 1/3 [■(-1&2&2@2&-1&2@2&2&-1)] is orthogonal.
03:50 (ii)Verify that the matrix [■(1/√3&1/√6&-1/√2@1/√3&-2/√6&0@1/√3&1/√6&1/√2)] is orthogonal.
07:59 Q2.(i) Show that the matrix 1/√2 [■(1&i@-i&-1)] is unitary.
10:42 (ii) Prove that the matrix [■((1+i)/2&(-1+i)/2@(1+i)/2&(1-i)/2)] is unitary.
14:55 Q3.(i) Let A = [■(2&1&-1@-1&2&1@3&4&1)], find A+A’ and A-A’ and hence express A as the sum of a symmetric and a skew symmetric matrix.
17:50 (ii) Write the following matrix as the sum of a symmetric and a skew-symmetric matrix [■(1&2&3@4&5&6@7&0&0)].
20:20 Q4.(i) Show that the matrix [■(0&6&7@-6&0&8@-7&-8&0)] is skew-symmetric.
21:41 (ii)Show that the matrix A, where A= [■(1/6&(-2)/6&1/6@(-2)/6&4/6&(-2)/6@1/6&(-2)/6&1/6)] is symmetric.
22:52 Q5.If A=[■(1&1-i&2@1+i&3&i@2&-i&0)], prove that A is a Hermitian matrix.
24:12 Q6.Prove that the matrix A^2 is symmetric if either A is symmetric or A is skew-symmetric.
26:24 Q7.If A be any square matrix then show that A+A’ is symmetric and A-A’ is skew-symmetric.
28:24 Q8.If A is skew – symmetric matrix, then show that iA is Hermitian.
30:13 Q9.If A,B are symmetric matrices of same order, then so is called A+B.
30:58 Q10.Show that the matrix B^θ AB is Hermitian or skew-Hermitian according as A is Hermitian or skew-Hermitian.
33:59 Q11.Show that all positive integral power of a symmetric matrix are symmetric.
35:19 Q12.If A and B are symmetric matrices of order n , then show that AB + BA is symmetric and AB-BA is skew-symmetric.
37:56 Q13.If A be any square matrix, prove that A+A^θ, AA^θ, A^θ A are all Hermitian and A-A^θ is skew-Hermitian.
40:26 Q14.Show that every square matrix is uniquely expressible as the sum of a Hermitian matrix and a skew-Hermitian matrix.
41:18 Q15.Show that every square matrix A can be uniquely expressed as P+iQ where P and Q are Hermitian matrices.
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Matrices | Bsc 2nd Semester | BA Bsc | Engineering Mathematics | Lecture -1 | Krishna Publication