Question

Determine which of the following matrices are (1) symmetric, (ii) singular, (iii) strictly diagonally dominant, (iv) positive definite. 2 0 0 1 3 0 0 0 4 symmetric [Choose] Singular [Choose] strictly diagonally dominant Yes Yes positive definite matrix [Choose]

Answer 1

Let A= 12 I 01 o 3. o . lo o 4 i) The transpose of the matrix A is A = 12.00n 1.3 0 Thus ATZA So A. is not symmetric. 1) A matrix is said to be singular if its determinant is zero. Here determinant A =14= 12.01 03 01 = 2.(12-09-1.0 +0 = 24(0). Á is non-zero, so Thus the determinant of A is not singular,

e 1) A square matrix is said to be strictly diagonally dominant if for every row of the matrix, the magnitude of the diagonal entry in a row is larger than the sum of other (non-diagonal) entries in that row. That is if A= (ai;) nxn is an nxn matrix then A is strictly diagonally dominant if Tail > [ laijl , for all i = 1,2,.., h. #j where, aij and denotes the entry in the j th column ith row) Here A= 12 Ion lo 3 ol lo o 4 ) since, 121 > 111 + 101 131 > 101 +101 141 > 101 +101 so A is strietly diagonally dominant.

iv) - A square matrix A is said to be positive definite if and only if all of its principal minors are positive. I alt 0 12 -- ain Let A ani azaroan aan lani an 2 no.. ann and let 10 11 = 10111, 1021= nam 012 , 1031 = I on a 12 C43 . jail 642-..ain ari a22 a23 liroog Dolazı A22 --- azn pasi as2 a33 | Tani ana... anul - -- - -- are all principal minors of A: tive The A is positive definite if and only if 1 Dil >0, 102170, 103170,---, 1 Dul >0.

Now for A= 12 I on lo o 4) 1011=270, 1021 = 13 s = 6-0 = 670, ID213 12 0 1 1D31= w lo 3 o = 24 >0. lo 04 o Hence, the matrix A is positive definite. 1 0 1 Thus the given matrix A= 12 1o 3 lo o O 4 ) is not symmetric. not singular (non-singular) Strictly diagonally dominant. nu positive definite matrix