Page 42 - 11-Math-3 Matrices and Determinants
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Triangular Matrix: A square matrix A is named as triangular whether it is upper triangular
or lower triangular. For example, the matrices
1 2 3 1 0 0 0
0 1 4  2 0 0 are triangular matrices of order 3 and 4
0 0 6 and  3 1 5 0
2 3 1
4
-1
respectively. The first matrix is upper triangular while the second is lower triangular.
Note: Diagonal matrices are both upper triangular and lower triangular.
Symmetric Matrix: A square matrices A = [aij]n x n is called symmetric if At = A.
From At = A, it follows that [a’ij]n x n = [aij]n x n
which implies that a’ij = aji for i, j = 1, 2, 3, ......., n.
but by the definition of transpose, a’ij = aji for i, j = 1, 2, 3, ......., n.
Thus aij = aji for i, j = 1, 2, 3, ......., n.
and we conclude that a square matrix A = [aij]n x n is symmetric if aij = aji.
For example, the matrices
a h g 1 3 2 -1
b  0 5 
1 3 ,  h f f  and  3 5 1 6  are symmetric.
3 2   6 -2
 g c  2 -2
-1 
3 
Skew Symmetric Matrix : A square matrix A = [aij]n x n is called skew symmetric or anti-
symmetric if At = -A.
From At = -A, it follows that [a’ij] = for i, j = 1, 2, 3, ......., n
which implies that a’ij = -aij for i, j = 1, 2, 3, ......., n
but by the definition of transpose a’ij = aji for i, j = 1, 2, 3,..., n
Thus -aij = aji or aij = -aji
Alternatively we can say that a square matrix A = [aij]nxn is anti-symmetric
if aij = -aji .
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