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Problem: Is matrix [[-5,-8,0],[3,5,0],[1,2,-1]] [ Calculator, Method and examples ]

Solution:
Your problem -> Is matrix [[-5,-8,0],[3,5,0],[1,2,-1]]

A matrix, in which number of rows and number of columns are equal, is called a square matrix.

A =
 -5 -8 0 3 5 0 1 2 -1

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.

A =
 -5 -8 0 3 5 0 1 2 -1

|A| =
 -5 -8 0 3 5 0 1 2 -1

=
-5 ×
 5 0 2 -1
+8 ×
 3 0 1 -1
+0 ×
 3 5 1 2

=-5 xx (5 × (-1) - 0 × 2) +8 xx (3 × (-1) - 0 × 1) +0 xx (3 × 2 - 5 × 1)

=-5 xx (-5 +0) +8 xx (-3 +0) +0 xx (6 -5)

=-5 xx (-5) +8 xx (-3) +0 xx (1)

= 25 -24 +0

=1

Here, |A| != 0, so A is nonsingular matrix

A square matrix A is called an involutary matrix, if A^2 = I where I is the identity matrix.

A =
 -5 -8 0 3 5 0 1 2 -1

A×A=
 -5 -8 0 3 5 0 1 2 -1
×
 -5 -8 0 3 5 0 1 2 -1

=
 -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 0 0 0 1 0 0 0 1

Here A^2 = I, so A is an involutary matrix

A square matrix A is called a periodic matrix, if A^m = A for some positive integer m.

A =
 -5 -8 0 3 5 0 1 2 -1

A×A=
 -5 -8 0 3 5 0 1 2 -1
×
 -5 -8 0 3 5 0 1 2 -1

=
 -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 0 0 0 1 0 0 0 1

(A^2)×A=
 1 0 0 0 1 0 0 0 1
×
 -5 -8 0 3 5 0 1 2 -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

=
 -5 -8 0 3 5 0 1 2 -1

Here A^3 = A, so A is a periodic matrix of period 2

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