is Orthogonal Matrix Example-2

1. is Orthogonal Matrix ?
`[[1,0],[0,-1]]`

Solution:

A square matrix `A`, such that `A xx A^T = I`, is called an orthogonal matrix, where `I` is an identity matrix and `A^T` is the transpose of matrix `A`.

`A`

	`1`	`0`
	`0`	`-1`

`A^T`

	`1`	`0`
	`0`	`-1`

	`1`	`0`
	`0`	`-1`

`A×(A^T)`

	`1`	`0`
	`0`	`-1`

	`1`	`0`
	`0`	`-1`

	`1×1+0×0`	`1×0+0×-1`
	`0×1-1×0`	`0×0-1×-1`

	`1+0`	`0+0`
	`0+0`	`0+1`

	`1`	`0`
	`0`	`1`

`A xx A^T = I`, So `A` is an orthogonal matrix

2. is Orthogonal Matrix ?
`[[1,2,3],[4,5,6],[7,8,9]]`

Solution:

A square matrix `A`, such that `A xx A^T = I`, is called an orthogonal matrix, where `I` is an identity matrix and `A^T` is the transpose of matrix `A`.

`A`

`1`	`2`	`3`
`4`	`5`	`6`
`7`	`8`	`9`

`A^T`

`1`	`2`	`3`
`4`	`5`	`6`
`7`	`8`	`9`

`1`	`4`	`7`
`2`	`5`	`8`
`3`	`6`	`9`

`A×(A^T)`

`1`	`2`	`3`
`4`	`5`	`6`
`7`	`8`	`9`

`1`	`4`	`7`
`2`	`5`	`8`
`3`	`6`	`9`

`1×1+2×2+3×3`	`1×4+2×5+3×6`	`1×7+2×8+3×9`
`4×1+5×2+6×3`	`4×4+5×5+6×6`	`4×7+5×8+6×9`
`7×1+8×2+9×3`	`7×4+8×5+9×6`	`7×7+8×8+9×9`

`1+4+9`	`4+10+18`	`7+16+27`
`4+10+18`	`16+25+36`	`28+40+54`
`7+16+27`	`28+40+54`	`49+64+81`

`14`	`32`	`50`
`32`	`77`	`122`
`50`	`122`	`194`

`A xx A^T != I`, So `A` is not an orthogonal matrix

This material is intended as a summary. Use your textbook for detail explanation.
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2. Example-2