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MATRICES EXAMPLES
Example 1: Forming a matrix from equations
Consider the simultaneous equations:
4x + 5y + 3z
=
0
3x + 14y + 7z
=
0
10x + 13y + 8z
=
0
can be written as a matrix as
|||||
4
5
3
3
14
7
10
13
8
|||||.
This matrix is multiplied by the column vector, {x y z}T, and will be equal to the column vector, {0 0 0}T, which represents the right hand side of the simultaneous equations. This is a 3 × 3 square matrix.
Example 2: Identification of matrix properties.
Consider the matrices:
|A| = |||||
1
0
1
0
1
0
1
0
1
|||||
|B| = |||||
1
5
3
0
14
7
0
0
8
|||||
|C| = |||||
5
3
14
7
0
8
|||||
|D| = |||||
1
0
0
1
1
0
1
1
1
|||||
-
Which matrix (matrices), if any, are square?
-
Which matrix(matrices), if any, are diagonal matrices?
-
Which matrix(matrices), if any, are symmetric matrices?
-
Which matrix(matrices), if any, are lower diagonal matrices?
-
Which matrix(matrices), if any, are upper diagonal matrices?
-
Which matrix (matrices), if any, are square? A, B, D; all of these are 3 × 3; matrix C is a 2 × 3
-
Which matrix(matrices), if any, are diagonal matrices? none; all of them have elements not on the diagonal
-
Which matrix(matrices), if any, are symmetric matrices? A is symmetric since all elements aij = aji
-
Which matrix(matrices), if any, are lower diagonal matrices? D; all elements above the diagonal are zero
- Which matrix(matrices), if any, are upper diagonal matrices? none; all matrices have nonzero elements below the diagonal