Consider a matrix A=[ matrix -1&1&1\\ 1&-1&1\\ 1&1&-1 matrix ].if|2 ad 2 adi(3 ad mathcal A ^ -1 ))|=2^ a .3^ b , then the value of a - 2b is

Question

Consider a matrix  A = \(\begin{bmatrix}-1&1&1\\1&-1&1\\1&1&-1\end{bmatrix}\). If |2 adj(3 adj(4A^-1))| = 2^a.3^b, then the value of a - 2b is 

Answer 1

A = \(\begin{bmatrix}-1&1&1\\1&-1&1\\1&1&-1\end{bmatrix}\)

|A| = -1 + 1 + 1 + 1 + 1 + 1 = 4

Now, 

|2 adj (3adj(4A^-1))| = 2³|adj(3adj (4A^-1))|

(\(\because\) |KA| = Kⁿ|A|)

 = 2³|3adj(4A^-1)|² (\(\because\) |adj A| = |A|^n-1)

 = 2³(3³|adj (4A^-1)|)² (\(\because\) |KA| = Kⁿ|A|)

= 2³.3⁶ |adj(4A^-1)|⁴

= 2³.3⁶(4³. (A^-1))⁴ (\(\because\) |KA| = Kⁿ|A|)

=  2³.3⁶(4³. \(\frac14\)) (\(\because\) |A^-1| = \(\frac1{|A|}=\frac14\))

= 2³.3⁶(4²)⁴

= 2³.3⁶.4⁸

= 2³.3⁶.2¹⁶

= 2¹⁹.3⁶

\(\therefore\) a = 19, b = 6

\(\therefore\) a - 2b = 19 - 12 = 7