Question 2 and 3, Review Exercise

Solutions of Question 2 and 3 of Review Exercise of Unit 02: Matrices and Determinants. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan.

For the matrix $A=\left[\begin{array}{ccc}1 & 2 & 0 \\ -3 & 4 & 9 \\ 2 & 1 & 6\end{array}\right]$; find $A_{13}, A_{23}$ and $A_{33}$; hence find $|A|$.

Solution.

Given: $$\begin{align*} A&=\left[\begin{array}{ccc}1 & 2 & 0 \\ -3 & 4 & 9 \\ 2 & 1 & 6\end{array}\right]\\ A_{13} &= (-1)^{1+3} \left| \begin{array}{cc} -3 & 4 \\ 2 & 1 \end{array} \right| = -3 - 8 = -11\\ A_{23} &= (-1)^{2+3} \left| \begin{array}{cc} 1 & 2 \\ 2 & 1 \end{array} \right| = -(1 - 4) = 3\\ A_{33} &= (-1)^{3+3} \left| \begin{array}{cc} 1 & 2 \\ -3 & 4 \end{array} \right| = 4 +6 = 10\\ |A|&= 1(24-9)-2(-18-18)+0\\ &=15+72\\ &=87 \end{align*}$$

Prove that if $A^{-1}=A^{t}$ then $\left|A A^{t}\right|=1$.

Solution.

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