1. Auto detect the matrix type ?
[-5-8035012-1]Solution:
A matrix, in which number of rows and number of columns are equal, is called a square matrix.
The number of rows(3) and number of columns(3) are equal, So
A is a square matrix
A square matrix A, such that |A|≠0, is called nonsingular matrix.
=-5×(5×(-1)-0×2)+8×(3×(-1)-0×1)+0×(3×2-5×1)=-5×(-5+0)+8×(-3+0)+0×(6-5)=-5×(-5)+8×(-3)+0×(1)=25-24+0=1Here,
|A|≠0, so
A is nonsingular matrix
A square matrix A is called an involutary matrix, if A2=I where I is the identity matrix.
= | | -5×-5-8×3+0×1 | -5×-8-8×5+0×2 | -5×0-8×0+0×-1 | | | 3×-5+5×3+0×1 | 3×-8+5×5+0×2 | 3×0+5×0+0×-1 | | | 1×-5+2×3-1×1 | 1×-8+2×5-1×2 | 1×0+2×0-1×-1 | |
|
= | | 25-24+0 | 40-40+0 | 0+0+0 | | | -15+15+0 | -24+25+0 | 0+0+0 | | | -5+6-1 | -8+10-2 | 0+0+1 | |
|
Here
A2=I, so
A is an involutary matrix
A square matrix A is called a periodic matrix, if Am=A for some positive integer m.
= | | -5×-5-8×3+0×1 | -5×-8-8×5+0×2 | -5×0-8×0+0×-1 | | | 3×-5+5×3+0×1 | 3×-8+5×5+0×2 | 3×0+5×0+0×-1 | | | 1×-5+2×3-1×1 | 1×-8+2×5-1×2 | 1×0+2×0-1×-1 | |
|
= | | 25-24+0 | 40-40+0 | 0+0+0 | | | -15+15+0 | -24+25+0 | 0+0+0 | | | -5+6-1 | -8+10-2 | 0+0+1 | |
|
= | | 1×-5+0×3+0×1 | 1×-8+0×5+0×2 | 1×0+0×0+0×-1 | | | 0×-5+1×3+0×1 | 0×-8+1×5+0×2 | 0×0+1×0+0×-1 | | | 0×-5+0×3+1×1 | 0×-8+0×5+1×2 | 0×0+0×0+1×-1 | |
|
= | | -5+0+0 | -8+0+0 | 0+0+0 | | | 0+3+0 | 0+5+0 | 0+0+0 | | | 0+0+1 | 0+0+2 | 0+0-1 | |
|
Here
A3=A, so
A is a periodic matrix of period 2
This material is intended as a summary. Use your textbook for detail explanation.
Any bug, improvement, feedback then