`|[(log)_3 512,(log)_4 3],[(log)_3 8,(log)_4 9]|xx|[(log)_2 3,(log)_8 3],[(log)_3 4,(log)_3 4]|=` (a) 7 (b) 10 (c) 13 (d) 17

Options

A. 7
B. 10
C. 13
D. 17

Answer

B

Explanation

`|(log_(3) 512,log_(4) 3),(log_(3) 8,log_(4) 9)|xx|(log_(2) 3,"log"_(8) 3),("log"_(3) 4,log_(3)4 )|`
`= |(log_(3) 2^(9)" " log_(2^(2)) 3 ),(log_(3) 2^(3) " "log_(2) 3^(2))|xx|(log_(2) 3" "log_(2^(3)) 3),(log_(3) 2^(2) " "log_(3) 2^(2))|`
`|(9 log_(3) 2,(1)/(2) log_(2) 3),(3 log_(3) 2,(2)/(2) log_(2) 3)|xx|(log_(2) 3,(1)/(3) log_(2) 3),(2 log_(3) 2,2 log_(3) 2)|`
`= (9 -(3)/(2)) xx (2- (2)/(3)) = (15)/(2) xx (4)/(3) = 10 [ :' log_(2) 3 xx log_(3) 2 = 1]`
Thus, the determinant of a square matrix of order 2 is equal to the product of the diagondal elements minus the product of off-diagonal elements.
if `A = [(a_(11),a_(12),a_(13)),(a_(21),a_(22),a_(23)),(a_(31),a_(32),a_(33))]` is a square matrix of order 3, then the expression
`a_(11) a_(22) a_(33) + a_(12) a_(23) a_(31) + a_(13) a_(32) a_(21)`
`- a_(11) a_(23) a_(32) - a_(22) a_(13) a_(31) - a_(21) a_(21) a_(33)`
is defined as the determinant of A
i.e., `|A| = |(a_(11),a_(12),a_(13)),(a_(21),a_(22),a_(23)),(a_(31),a_(32),a_(33))|`
or, `|A| = a_(11) a_(22) a_(33) + a_(12) a_(23) a_(31) + a_(13) a_(32) a_(21)`
`- a_(11) a_(23) a_(32) - a_(22) a_(31) a_(13) - a_(33) a_(12) a_(21)`...(ii)
or, `|A| = |(a_(11),a_(12),a_(13)),(a_(21),a_(22),a_(23)),(a_(31),a_(32),a_(33))|`
`= a_(11) (a_(22) a_(33) - a_(23) a_(32)) - a_(12) (a_(33) a_(21) - a_(23) a_(31)) + a_(13) (a_(32) a_(21) - a_(22) a_(31))`
`= a_(11) |(a_(22),a_(23)),(a_(32),a_(33))| -a_(21) |(a_(21),a_(23)),(a_(31),a_(33))| + a_(13) |(a_(21),a_(22)),(a_(31),a_(32))|` [Using (i)]
`= (-1)^(1 +1) a_(11) |(a_(22),a_(23)),(a_(32),a_(33))| + (-1)^(1 +2) a_(12) |(a_(21),a_(23)),(a_(31),a_(33))| + (-1)^(1 +3) a_(13) |(a_(21),a_(22)),(a_(31),a_(32))|`
Thus, the determinant of a square matrix of order 3 is the sum of the product of elements `a_(1j)` in first row with `(-1)^(1 +j)` times the determinant of a `2 xx 2` sub-matrix obtained by leaving the first row and column passing through the element.
The above expansion of `|A|` is known as the expansion along first row.
There are three rows the three column in a square matrix of order 3. The expression (ii) for the determinant of a square matrix of roder 3 can be arranged in various forms to obtain the expansion of `|A|` along any of its rows or columns. Infact, to expand `|A|` about a row or a column we multiple each element `a_(ij) " in " i^(th)` row with `(-1)^( i +j)` times the determinant of the sub matrix obtained by leaving the row and column passing through the element.