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Let matrix A=[[x,y,-z],[1,2,3],[1,1,2]] ...

Let matrix `A=[[x,y,-z],[1,2,3],[1,1,2]]` where `x,y,zepsilonN`. If `det.(adj(adj.A))=2^8.(3^4)` then the number of such matrices A is :

Text Solution

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Here, `|adj(adjA)| = 2^8*3^4`
We know,` |adj(adjA)| = |A|^((n-1)^2)`, where `n` is order of the matrix.
Here, `n = 3`
`:. |A|^(2^2) = 2^8*3^4`
`=>|A| = 2^2*3 = 12`
Now, `A = [[x,y,-z],[1,2,3],[1,1,2]]`
`:. |A| = [x(4-3)-y(2-3)-z(1-2)] = x+y+z`
`:. x+y+z = 12`
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