let n be Element of NAT ; :: thesis: for K being Field
for A being Matrix of n,K st n > 0 & A * 1,1 <> 0. K holds
ex P, Q being Matrix of n,K st
( P is invertible & Q is invertible & ((P * A) * Q) * 1,1 = 1. K & ( for i being Element of NAT st 1 < i & i <= n holds
((P * A) * Q) * i,1 = 0. K ) & ( for j being Element of NAT st 1 < j & j <= n holds
((P * A) * Q) * 1,j = 0. K ) )
let K be Field; :: thesis: for A being Matrix of n,K st n > 0 & A * 1,1 <> 0. K holds
ex P, Q being Matrix of n,K st
( P is invertible & Q is invertible & ((P * A) * Q) * 1,1 = 1. K & ( for i being Element of NAT st 1 < i & i <= n holds
((P * A) * Q) * i,1 = 0. K ) & ( for j being Element of NAT st 1 < j & j <= n holds
((P * A) * Q) * 1,j = 0. K ) )
let A be Matrix of n,K; :: thesis: ( n > 0 & A * 1,1 <> 0. K implies ex P, Q being Matrix of n,K st
( P is invertible & Q is invertible & ((P * A) * Q) * 1,1 = 1. K & ( for i being Element of NAT st 1 < i & i <= n holds
((P * A) * Q) * i,1 = 0. K ) & ( for j being Element of NAT st 1 < j & j <= n holds
((P * A) * Q) * 1,j = 0. K ) ) )
assume A1: ( n > 0 & A * 1,1 <> 0. K ) ; :: thesis: ex P, Q being Matrix of n,K st
( P is invertible & Q is invertible & ((P * A) * Q) * 1,1 = 1. K & ( for i being Element of NAT st 1 < i & i <= n holds
((P * A) * Q) * i,1 = 0. K ) & ( for j being Element of NAT st 1 < j & j <= n holds
((P * A) * Q) * 1,j = 0. K ) )
then consider P being Matrix of n,K such that
A3: ( P is invertible & (P * A) * 1,1 = 1. K & ( for i being Element of NAT st 1 < i & i <= n holds
(P * A) * i,1 = 0. K ) & ( for j being Element of NAT st 1 < j & j <= n & A * 1,j = 0. K holds
(P * A) * 1,j = 0. K ) ) by AA3030;
consider Q being Matrix of n,K such that
A4: ( Q is invertible & ((P * A) * Q) * 1,1 = 1. K & ( for j being Element of NAT st 1 < j & j <= n holds
((P * A) * Q) * 1,j = 0. K ) & ( for i being Element of NAT st 1 < i & i <= n & (P * A) * i,1 = 0. K holds
((P * A) * Q) * i,1 = 0. K ) ) by A1, A3, AA3020;
for i being Element of NAT st 1 < i & i <= n holds
((P * A) * Q) * i,1 = 0. K

proof

let i be Element of NAT ; :: thesis: ( 1 < i & i <= n implies ((P * A) * Q) * i,1 = 0. K )
assume B1: ( 1 < i & i <= n ) ; :: thesis: ((P * A) * Q) * i,1 = 0. K
then (P * A) * i,1 = 0. K by A3;
hence ((P * A) * Q) * i,1 = 0. K by B1, A4; :: thesis: verum

end;

hence ex P, Q being Matrix of n,K st
( P is invertible & Q is invertible & ((P * A) * Q) * 1,1 = 1. K & ( for i being Element of NAT st 1 < i & i <= n holds
((P * A) * Q) * i,1 = 0. K ) & ( for j being Element of NAT st 1 < j & j <= n holds
((P * A) * Q) * 1,j = 0. K ) ) by A4, A3; :: thesis: verum