# Ch 03: Matrices and Determinants

• Fin $x$ and $y$ if $\left[ {\begin{array}{c} x+3&1\\ -3& 3y-4 \end{array}} \right]= \left[ {\begin{array}{c} 2&1\\ -3&2 \end{array}} \right]$ — BISE Gujrawala(2015)
• Solve for matrix $A$ if $\left[ {\begin{array}{c}4&3\\ 2&2 \end{array}} \right]A-\left[ {\begin{array}{c} 2&3\\ -1&-2 \end{array}} \right]= \left[ {\begin{array}{c} -1&-4\\ 3&6 \end{array}} \right]$ — BISE Gujrawala(2015)
• Prove without expansion $\left[ {\begin{array}{c} 6&7&8\\ 3&4&5\\ 2&3&4 \end{array}} \right]= 0$ — BISE Gujrawala(2015)
• Find inverse of $\left[ {\begin{array}{c} 2&1&0\\ 1&1&0\\ 2&-3&5 \end{array}} \right]$ — BISE Gujrawala(2017)
• Evaluate the determinant $\left[ {\begin{array}{c} 5&-2&5\\ 3&-1&4\\ -2&1&-2 \end{array}} \right]$ — BISE Gujrawala(2017)
• If $A=\left[ {\begin{array}{c} 1&2\\ a& b \end{array}} \right]$ and $A^2=\left[ {\begin{array}{c} 0&0\\ 0&0 \end{array}} \right]$ find the value of $a$ and $b$. — BISE Sargodha(2017), Gujrawala(2017)
• Find the value of $\lambda$ if $A=\left[ {\begin{array}{c} 4&\lambda&3\\ 7&3&6\\2&3&1 \end{array}} \right]$ is singular — BISE Sargodha(2015)
• Find the value of $x$, $\left[ {\begin{array}{c} 3&1&x\\ -1&3&4\\ x&1&0 \end{array}} \right]=0$ — BISE Sargodha(2015)
• Evaluate $\left[ {\begin{array}{c} a+l&a-l&a\\ a&a+l&a-l\\ a-l&a&a+l \end{array}} \right]$ — BISE Sargodha(2015)
• If $A=\left[ {\begin{array}{c} l\\1+i\\i \end{array}} \right]$ find $A(\bar A)^t$ — BISE Sargodha(2016)
• Find inverse of $\left[ {\begin{array}{c} 2i&i\\i&-i \end{array}} \right]$ — BISE Sargodha(2016)
• Use Cramer's rule to solve the system of equations: — BISE Sargodha(2016)

$${\begin{array}{c} 2x_1+x_2-x_3=-4\\x_1+x_2-2x_3=-4\\-x_1+2x_2-x_3=1 \end{array}}$$

• Use Cramer's rule to solve the system of equations — BISE Sargodha(2017)

$${\begin{array}{c} 2x_1-x_2+x_3=8\\x_1+2x_2+2x_3=6\\x_1-2x_2-x_3=1 \end{array}}$$

• If $A=\left[ {\begin{array}{c} l&-1\\a&b \end{array}} \right]$ and $A^2=\left[ {\begin{array}{c} l&0\\0&1 \end{array}} \right]$ find the value of $a$ and $b$ — BISE Sargodha(2017)
• Show that $\left[ {\begin{array}{c} b&-1&a\\a&b&0\\1&a&b \end{array}} \right]=a^3+b^3$ — BISE Sargodha(2017)
• $A=\left[ {\begin{array}{c} i&0\\i&-i \end{array}} \right]$, show that $A^4=I_2$ — BISE Lahore(2017)
• $A=\left[ {\begin{array}{c} i&l+i\\l&-i \end{array}} \right]$ show that $A-(\bar A)$ is skew-hermitian. — BISE Lahore(2017)
• Without expansion show that $\left[ {\begin{array}{c} ba&ca&ab\\ \frac{1}{a}&\frac{1}{b}&\frac{1}{c}\\a&b&c \end{array}} \right]=0$ — BISE Lahore(2017)
• Verify that $(AB)^t=B^t A^t$ if $A=\left[ {\begin{array}{c} 1&-1&2\\0&3&1 \end{array}} \right]$, $B=\left[ {\begin{array}{c} 1&1\\3&2\\0&-1 \end{array}} \right]$ — BISE Lahore(2017)
• Solve the following matrix equations for $A$.
$\left[ {\begin{array}{c} 4&3\\2&2 \end{array}} \right]A- \left[ {\begin{array}{c} 2&3\\-1&-2 \end{array}} \right]= \left[ {\begin{array}{c} -1&4\\3&6 \end{array}} \right]$— FBISE (2016)
• Solve the equation $\left[ {\begin{array}{c} x&0&1&1\\0&1&1&-1\\1&-2&3&4\\-2&x&1&-1 \end{array}} \right]=0$ — FBISE (2016)
• Use matrices to solve the following system — FBISE (2017) $${\begin{array}{c} x+y=2\\2x-z=1\\2y-3z=-1\end{array}}$$
• Without expansion verify that $\left[ {\begin{array}{c} -a&0&c\\0&a&-b\\b&-c&0 \end{array}} \right]=0$