Let A be a 3 × 3 real matrix. If det ( 2 Adj ( 2 Adj ( Adj ( 2A ) )...

∴ | KA | = {K}^{n} | A | ,$If$A$is$n \times n$matrix</p><p>$| Adj A | = | A{|}^{n - 1}$If$A$is$n \times n$matrix</p><p>$| 2 Adj ( 2 Adj ( Adj ( 2A ) ) ) | = {2}^{41}$</p><p>$⇒ {2}^{3} | Adj ( 2 Adj ( Adj ( 2A ) ) ) | = {2}^{41}$<br />$⇒ | Adj ( 2 Adj ( Adj ( 2A ) ) ) | = {2}^{38}$<br />$⇒ ( | 2 Adj ( Adj ( 2A ) ) | {)}^{2} = {2}^{38}$<br />$⇒ | 2 Adj ( Adj ( 2A ) ) | = {2}^{19}$<br />$⇒ {2}^{3} | Adj ( Adj ( 2A ) ) | = {2}^{19}$<br />$⇒ | Adj ( Adj ( 2A ) ) | = {2}^{16}$<br />$⇒ ( | Adj ( 2A ) | {)}^{2} = {2}^{16}$<br />$⇒ | Adj ( 2A ) | = {2}^{8}$<br />$⇒ | ( 2A ) {|}^{2} = {2}^{8}$<br />$⇒ | ( 2A ) | = {2}^{4}$<br />$⇒ {2}^{3} | A | = {2}^{4}$<br />$⇒ | A | = 2$</p><p>So, \left|{A}^{2}\right| = {\left|A\right|}^{2} = 4$