Matrices [Pearson Edexcel 4MB1 topic question (2014-2016)]

 Matrices

1	(Edexcel/Math B/2014/jan/paper01/q19) \( \mathbf{A}=\left(\begin{array}{rr}1 & x \\ -3x & 2\end{array}\right) \), \( \mathbf{B}=\left(\begin{array}{rr}-2 & 2x \\ -1 & 5\end{array}\right) \) (a) Write down and simplify the matrix $4A-3B$. [2] Given that \( \mathbf{4A-3B}=\left(\begin{array}{rr}10 & 4 \\ 27 & -7\end{array}\right) \) (b)calculate the value of $x$.[2]
2	(Edexcel/Math B/2014/jan/paper02/q4) (a) Find the inverse of the matrix \( \left(\begin{array}{rr}2 & -1 \\ 5 & -2\end{array}\right) \) Hence or otherwise find the value of x and the value of y such that \( \left(\begin{array}{rr} 2 & -1 \\ 5 & -2\end{array}\right) \)\(\left(\begin{array}{rr} x \\ y \end{array}\right) \)=\( \left(\begin{array}{rr} -1 \\ 1 \end{array}\right) \) [The inverse of matrix \( \left(\begin{array}{rr}a & b \\ c & d\end{array}\right) \) is $\frac {1}{ad-bc}$ \( \left(\begin{array}{rr}d & -b \\ -c & a\end{array}\right) \)]
3	(Edexcel/Math B/2014/janR/paper01/q14) (a) Given that $\mathbf{A}$ = \( \left(\begin{array}{rr}x & 0 \\ -x & -x\end{array}\right) \), find , in terms of $x,A^2$.....[2] (b) Given that $\mathbf{A}^2=9\mathbf{I}$, where $\mathbf{I}$=\( \left(\begin{array}{rr}1 & 0 \\ 0 & 1\end{array}\right) \), find a value of $x$....[1]
4	(Edexcel/Math B/2014/jan/paper01/q1) Given that \( \left(\begin{array}{rr}0 & -1 \\ -1 & 0\end{array}\right) \)\(\left(\begin{array}{rr}-1 \\ 0\end{array}\right) \)=\( \left(\begin{array}{rr}0 \\ -1 \end{array}\right) \) (a) Find the values of $x$....[3] (b) Hence find the possible values of $y$. ....[3]
5	(Edexcel/Math B/2014/juneR/paper01/q8) $\mathbf{A}$= \( \left(\begin{array}{rr}4 \\ -3 \end{array}\right) \) and B=(3 -1). Find the matrix product AB.....[2]
6	(Edexcel/Math B/2014/june/paper02R/q2) Given that \[ A = \begin{pmatrix} 3 & -2 \\ -1 & 4 \end{pmatrix}, \] find the \(2 \times 2\) matrix \(B\) such that \[ A + 2B = A^2. \]....[5]
7	(Edexcel/Math B/2014/jan/paper01/q1) d is the determinant of the matrix A. Given that \( \mathbf{A}=\left(\begin{array}{ll}4 x & 6 \\ 3 & 2\end{array}\right) \) (i) write down an expression for d in terms of x. (ii) Hence find the value of x for which d=2x. ....[2]
8	(Edexcel/Math B/2015/jan/paper01/q19) \[ \mathbf{A}=\left(\begin{array}{ll} 1 & \cdot \\ 3 & \cdot 4 \end{array}\right), \quad \mathbf{B}=\left(\begin{array}{ll} 3 & 2 \\ 1 & 1 \end{array}\right) \] Find \( \mathbf{A B}-\mathbf{B A} \).....[4]
9	(Edexcel/Math B/2015/june/paper01/q5) Find the value of the determinant of the matrix \( \left(\begin{array}{cc}3 . & 7 \\ 1 . & 5\end{array}\right) \).....[2]
10	(Edexcel/Math B/2015/june/paper01/q23) \[ \mathbf{A}=\left(\begin{array}{cc} x & 4-6 x \\ 6+3 y & 4 y \end{array}\right), \quad \mathbf{B}=\left(\begin{array}{cc} 2 x & 2-8 x \\ 7+4 y & -y \end{array}\right), \quad \mathbf{C}=\left(\begin{array}{ll} 4 & 10 \\ 3 & 19 \end{array}\right) \] Given that \( 4 \mathbf{A}-3 \mathbf{B}=\mathbf{C} \), calculate the value of \( x \) and the value of \( y \). ....[4]
11	(Edexcel/Math B/2015/june/paper02R/q1) \( \mathrm{A}, \mathrm{B} \) and C are three matrices. \[ \mathbf{A}=\binom{p \cdot 3}{-3 \cdot 4}, \mathbf{B}=\left(\begin{array}{rr} 3 & q \\ \cdot & -3 \\ -2 & -3 \end{array}\right) \text { and } \cdot \mathbf{C}=\left(\begin{array}{rrr} -12 & \cdot & r \\ r & \cdot & -24 \end{array}\right) \] Given that \( \mathbf{A B}=\mathbf{C} \), find the value of \( p \), the value of \( q \) and the value of \( r \)....[4]
12	(Edexcel/Math B/2016/jan/paper01/q18) \[ \mathbf{A}=\left(\begin{array}{ll} 1 & 2 \\ -2 & 1 \end{array}\right) \quad \cdot \quad \mathbf{B}=\left(\begin{array}{cc} -2 & \cdot 3 \\ 1 & \cdot 5 \end{array}\right) \] Find (a) \( 2 \mathrm{~A}-3 \mathrm{~B} \)...[2] (b) $\mathbf{AB}$.....[2]
	to be continued

Answer Key

1.   (a) $\left(\begin{array}{cc}10 & -2 x \\ -12 x+3 & -7\end{array}\right)$ (b) $x=-2$ 
2.   (a) $\left(\begin{array}{cc}-2 & 1 \\-5 & 2\end{array}\right)$ (b) $ x=3, y=7 \quad$ 
3.   (a) $\left(\begin{array}{cc}x^2 & 0 \\0 & x^2\end{array}\right) \quad$ (b) $ x=3$ (or) $-3 $
4.   (a) $x=2, x=-2 \quad$ (b) $y=8, y=4 \quad$ 
5.   $\left(\begin{array}{cc}12 & -4 \\ -9 & 3\end{array}\right)$
6.   $\left(\begin{array}{cc}4 & -6 \\ -3 & 7\end{array}\right)$
7.   (i) $8 x-18$ (ii) $x=3$
8.   $\left(\begin{array}{cc}-4 & -10 \\ 9 & 4\end{array}\right)$ 
9.   8
10.   $x=-2, y=1$ 
11.   $p=-2, q=4, r=-17 \quad$ 
12.   (a) $\left(\begin{array}{cc}8 & -5 \\ -7 & -13\end{array}\right)$ (b) $\left(\begin{array}{cc}0 & 13 \\ 5 & -1\end{array}\right) \quad$ 
13.   $x=\frac{7}{3}, x=-1 \quad$ 
14.   (a) $\frac{1}{7}\left(\begin{array}{ll}-1 & 2 \\ -5 & 3\end{array}\right)$ (b) $x=2, y=1$
15.   $\left(\begin{array}{cc}6 & -3 \\ -12 & 8\end{array}\right) $
16.   $a=4, b=3, c=36$
17.   $A B=\left(\begin{array}{cc}-8 & -1 \\ 14 & 2\end{array}\right), \lambda=4$
18.   (a) $\left(\begin{array}{cc}{1}&{-19} \\ {-13}{23}\end{array}\right)$ (b) $\left(\begin{array}{ccc}{-7}{1}{-11} \\ {17}{4}{19}\end{array}\right)$
19.   (a) $ x=3,-3 \quad$ (b) $ y=11, y=-1$