If A is a non-singular matrix, then (A^T)^–1 = _________.

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Solution

It is given that, A is a non-singular matrix.

∴ |A| ≠ 0

⇒ |A^T| ≠ 0 (|A| = |A^T|)

So, A^Tis also invertible.

Now,

$A A^{- 1} = I_{n} = A^{- 1} A$

$\Rightarrow {(A A^{- 1})}^{T} = {(I_{n})}^{T} = {(A^{- 1} A)}^{T}$

$\Rightarrow {(A^{- 1})}^{T} A^{T} = I_{n} = A^{T} {(A^{- 1})}^{T}$

$\Rightarrow {(A^{T})}^{- 1} = {(A^{- 1})}^{T}$ (Using definition of inverse of a matrix)

If A is a non-singular matrix, then (A^T)^–1 = ${(A^{- 1})}^{T}$ .

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If A is a non-singular matrix, then (AT)–1 = _________.

It is given that, A is a non-singular matrix. ∴ |A| ≠ 0 ⇒ |AT| ≠ 0 (|A| = |AT|) So, AT is also invertible. Now, AA-1=In=A-1A ⇒AA-1T=InT=A-1AT ⇒A-1TAT=In=ATA-1T ⇒AT-1=A-1T (Using definition of inverse of a matrix) If A is a non-singular matrix, then (AT)–1 = A-1T .

If A is a non-singular matrix, then (A^T)^–1 = _________.