Question

If matrix A =[2 3 4
d e f
g h i ]
and |A| = - 2 , find
[3g - 2 + d 4
3h - 3 + e 6
3i - 4 + f 8]
can someone give me a clue on how to do it

Answer 1

find the determinant ...

Answer 2

-(4ge+2fh+3di)
2 3 4 2 3
d e f d e
g h i g h
+(2ei+3fg+4dh)

therefore

(2ei+3fg+4dh) - (4ge+2fh+3di) = -2

2ei +3fg +4dh - 4ge -2fh -3di = -2

gives us some sort of a constraint to play with

Answer 3

can you clarify the second part of the setup? d4, e6, and f8 what do these represent? exponents, coefficients? or is there something missing?

Answer 4

it another matrix [ 3g -2+d 4
3h -3+e 6
3i -4+f 8]

Answer 5

ah, ok. and i assume theres nothing constraining us on the other matrix? or is it just using some parts of the first?

Answer 6

i wonder if we can do some row operations and transform one into the other

Answer 7

are you supposed to find the determinate of this second matrix?
if so, the det of a transposed matrix is the same
if you multiply a row (or column) by n, the det is multiplied by n

Answer 8

yes i am supposed to find determinate for second matrix based on first matrix determinent

Answer 9

step 1
if we start with the first matrix, and take its transpose,
this new matrix has the same det = -2

Answer 10

now we have a full detailed question to work with :)

Answer 11

if you multiply the first column by 3, you get the first column of the "target matrix"
similarly, if you multiply the last column by 2, you get the last column of the target matrix.

for the middle column,
det { a + A b
c +C d } (the first column is the sum of vector [a c]^T and [A C]^T )
is
det (a b
c d) + det(A b
C d)

Answer 12

that last bit is a 2x2 example. For your matrix, break the problem into the det of 2 matrices (with different 2nd columns)

Answer 13

is that enough info ? can you make progress?

Answer 14

one of the matrices will have the second column be a multiple of the 3rd column. This means its det is 0

that leaves you with the det of the other matrix...

Answer 15

tq phi