a Show that A is invertible and that rankA 3 b Show in gene

a) Show that A is invertible and that rank(A) = 3

b) Show in general that if A is an n x n invertible matrix then rank(A) = n

a) To show that A is invertible, we can show that the determinant of A is non-zero

det(A) = 1(8-7) + 2(1+16) -3(-14-1)

=> 1 + 34 - 45

=> -10

Since det(A) is a non-zero number, hence all the columns of the matrix are linearly independent

Hence rank of matrix A = number of independent columns = 3

Hence proved

If A is an nXn invertible matrix that means det(A) is a non-zero number

AA^(-1) = I(nXn)

Since A is of order nXn, A^(-1) will also be of order nXn, hence there exists a matrix which maps A to the set of n independent columns

Hene the rank of matrix A = number of independents columns in matrix A = n