Solve Invertible Matrix Problem: Find Equivalent Conditions to "A is Invertible

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I have to write all possible equivalent conditions to "A is invertible," where A is an nxn matrix. can anyone help me out with this question
 
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Name one. You aren't trying very hard.
 
i don't understand the question
 
well you know that A^-1 = 1/det|A| *adj(A)

so for this to exist...det|A| can't be zero...and think of row-echelon form when finding A^-1

in row-echelon form, to get A^-1. how many non-zero rows must it have?How many pivot positions must it have?
 
Here's another one. A is invertible if there is a matrix B such that A*B=I. You are way behind. How about a statement in terms of the dimension of the kernel? How about expressing invertability in terms of the solutions to a system of linear equations? There's a lot of ways to express this concept.
 
A is also invertible when the determinant does not equal to 0
 

Thread 'Prove that the integral is equal to ##\pi^2/8##'

Prove $$\int\limits_0^{\sqrt2/4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx = \frac{\pi^2}{8}.$$ Let $$I = \int\limits_0^{\sqrt 2 / 4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx. \tag{1}$$ The representation integral of ##\arcsin## is $$\arcsin u = \int\limits_{0}^{1} \frac{\mathrm dt}{\sqrt{1-t^2}}, \qquad 0 \leqslant u \leqslant 1.$$ Plugging identity above into ##(1)## with ##u...
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