(a) Given

\(A=\begin{pmatrix}1&2\\0&-1\end{pmatrix},\quad B=\begin{pmatrix}1&-4\\2&5\\3&1\end{pmatrix},\quad C=\begin{pmatrix}3&-2&0\end{pmatrix},\)

write down all the possible products that can be formed by multiplying two of these matrices.

(b) Given

\(X=\begin{pmatrix}2&-2\\5&3\end{pmatrix},\qquad Y=\begin{pmatrix}4&1\\2&0\end{pmatrix},\)

(i) find \(X^{-1}\),

(ii) solve the matrix equation \(XZ=Y\).

(a) \(A=\begin{pmatrix}x+3&-x\\2x&x-3\end{pmatrix}\). Given that \(A\) does not have an inverse, find the exact values of \(x\).

(b) \(B=\begin{pmatrix}0&3\\-4&1\\5&2\end{pmatrix}\) and \(C=\begin{pmatrix}0&1&2\\3&-4&5\end{pmatrix}\).

(i) Write down the order of matrix \(B\).

(ii) Explain why \(CB\ne BC\).

(a) State the order of the matrix \(\begin{pmatrix}0&1&4&8\\5&8&1&6\end{pmatrix}\).

(b) The matrix \(A\) is given by \(A=\begin{pmatrix}2&-4\\-1&3\end{pmatrix}\).

(i) Find \(A^{-1}\).

(ii) Hence, given that \(ABA=I\), find \(B\).

(a) Given that \(A=\begin{pmatrix}2&5\\0&-1\\6&4\end{pmatrix}\) and that \(A+O=A\),

(i) state the order of the matrix \(A\),

(ii) write down the matrix \(O\).

(b) \(B=\begin{pmatrix}1&-1\\3&2\end{pmatrix}\) and \(C=\begin{pmatrix}0.4&0.2\\-0.6&0.2\end{pmatrix}\). Find the matrix product \(BC\) and state a relationship between \(B\) and \(C\).

(c) \(D=\begin{pmatrix}a&4a\\-1&5\end{pmatrix}\), where \(a\) is a positive integer. Find \(D^{-1}\) in terms of \(a\).

(a) Five teams took part in a competition in which each team played each of the other \(4\) teams. The following table represents the results after all the matches had been played.

Points in the competition were awarded to the teams as follows:

\(4\text{ for each match won},\qquad 2\text{ for each match drawn},\qquad 0\text{ for each match lost}.\)

(i) Write down two matrices whose product under matrix multiplication will give the total number of points awarded to each team.

(ii) Evaluate the matrix product from part (i) and hence state which team was awarded the most points.

(b) It is given that

\(A=\begin{pmatrix}1&-1\\2&4\end{pmatrix},\qquad B=\begin{pmatrix}5&0\\1&-2\end{pmatrix}.\)

(i) Find \(A^{-1}\).

(ii) Hence find the matrix \(C\) such that \(AC=B\).

It is given that

\(A=\begin{pmatrix}5&2\\4&-1\end{pmatrix}.\)

(i) Find \(A^{-1}\).

(ii) Hence find, in radians, the acute angles \(x\) and \(y\) such that

\(5\tan x+2\tan y=12,\)

\(4\tan x-\tan y=7.\)

Given that

\(A=\begin{pmatrix}5&2\\-9&-3\end{pmatrix},\qquad B=\begin{pmatrix}2&1\\6&5\end{pmatrix},\)

find

(i) \(A^{-1}\),

(ii) \(B^2\),

(iii) the matrix \(C\), where \(B^{-1}C+A=B\),

(iv) the matrix \(D\), where \(B^{-2}DA=I\).

(a) Find the values of \(a\) for which the determinant of \(\begin{pmatrix}2a&1\\4a&a\end{pmatrix}\) is \(6-3a\).

(b) Let \(A=\begin{pmatrix}2&1\\3&4\end{pmatrix}\) and \(B=\begin{pmatrix}2&0\\-3&5\end{pmatrix}\).

(i) Find \(A^{-1}\).

(ii) Hence find the matrix \(C\) such that \(AC=B\).

(c) Find the \(2\times2\) matrix \(D\) such that \(4D+3I=O\), where \(I\) is the identity matrix and \(O\) is the zero matrix.

(i) Find the inverse of the matrix

\(\begin{pmatrix}4&-2\\-5&3\end{pmatrix}.\)

(ii) Hence solve the simultaneous equations

\(8x-4y-5=0,\qquad -10x+6y-7=0.\)

(i) Find the inverse of the matrix

\(\begin{pmatrix}4&-2\\-5&3\end{pmatrix}.\)

(ii) Hence solve the simultaneous equations

\(8x-4y-5=0,\qquad -10x+6y-7=0.\)

(a) Given

\(A=\begin{pmatrix}2&-1\\1&-3\end{pmatrix}, \qquad B=\begin{pmatrix}0&-2\\3&-5\end{pmatrix},\)

find \((BA)^{-1}\).

(b) The matrix \(X\) is such that \(XC=D\), where

\(C=\begin{pmatrix}-2&5&3\\0&10&4\end{pmatrix}, \qquad D=\begin{pmatrix}-4&5&4\end{pmatrix}.\)

(i) State the order of the matrix \(C\).

(ii) Find the matrix \(X\).

\(A=\begin{pmatrix}a&3\\4&a+4\end{pmatrix}.\)

(i) Find the values of the constant \(a\) for which \(A^{-1}\) does not exist.

(ii) Given that \(a=4\), find \(A^{-1}\).

(iii) Hence find the matrix \(B\) such that \(AB=\begin{pmatrix}2&3\\4&-5\end{pmatrix}\).

\(A=\begin{pmatrix}3&2\\-1&1\end{pmatrix}.\)

(i) Find \(A^2\).

(ii) Find constants \(p\) and \(q\) such that

\(pA^2+qA=I.\)

Given that

\(A=\begin{pmatrix}2&3\\1&4\end{pmatrix} \quad\text{and}\quad B=\begin{pmatrix}1&4\\-2&5\end{pmatrix},\)

find

(i) \(A^{-1}\),

(ii) the matrix \(C\) such that \(CA=B\),

(iii) the matrix \(D\) such that \(A^{-1}D+B=I\).

(a) Given

\(A=\begin{pmatrix}3&1\\-1&2\\4&5\end{pmatrix},\qquad B=\begin{pmatrix}1&-2\\3&0\end{pmatrix}, \qquad C=AB,\)

(i) state the order of \(A\), (ii) find \(C\).

(b) The matrix

\(X=\begin{pmatrix}5&-12\\4&-7\end{pmatrix}.\)

(i) Find \(X^{-1}\).

(ii) Using \(X^{-1}\), find the coordinates of the point of intersection of the lines \(12y=5x-26\) and \(7y=4x-52\).

Given that

\(A=\begin{pmatrix}3&1\\2&4\end{pmatrix},\quad B=\begin{pmatrix}5&1\\2&4\\-1&0\end{pmatrix}, \quad\text{and}\quad C=\begin{pmatrix}-5&2\\3&1\end{pmatrix},\)

find

(a)(i) \(A+3C\),

(a)(ii) \(BA\).

(b)(i) Given that

\(X=\begin{pmatrix}1&-3\\4&-2\end{pmatrix},\)

find \(X^{-1}\).

(b)(ii) Hence find \(Y\), such that

\(XY=\begin{pmatrix}5&-10\\15&20\end{pmatrix}.\)

Four cinemas, \(P\), \(Q\), \(R\) and \(S\) each sell adult, student and child tickets. The number of tickets sold by each cinema on one weekday were

(i) Given that \(L=(1\ 1\ 1\ 1)\), construct a matrix \(M\), of the number of tickets sold, such that the matrix product \(LM\) can be found.

(ii) Find the matrix product \(LM\).

(iii) State what information is represented by the matrix product \(LM\).

An adult ticket costs \(\$5\), a student ticket costs \(\$4\) and a child ticket costs \(\$3\).

(iv) Construct a matrix \(N\), of the ticket costs, such that the matrix product \(LMN\) can be found and state what information is represented by the matrix product \(LMN\).

(a) It is given that \(A=\begin{pmatrix}4&-1\\a&b\end{pmatrix}\), \(B=\begin{pmatrix}2&3\\-5&4\end{pmatrix}\), and \(AB=\begin{pmatrix}13&8\\18&4\end{pmatrix}\). Find the value of \(a\) and of \(b\).

(b) It is given that \(X=\begin{pmatrix}3&-5\\-4&1\end{pmatrix}\), \(Y=\begin{pmatrix}-1&2\\4&0\end{pmatrix}\), and \(XZ=Y\).

(i) Find \(X^{-1}\).

(ii) Hence find \(Z\).

It is given that \(M=\begin{pmatrix}2&p\\-3&q\end{pmatrix}\), where \(p\) and \(q\) are integers.

(i) If \(\det M=13\), find an equation connecting \(p\) and \(q\).

(ii) Given also that \(M^2=\begin{pmatrix}4-3p&12\\-6-3q&-3p+q^2\end{pmatrix}\), find a second equation connecting \(p\) and \(q\).

(iii) Find the value of \(p\) and of \(q\).

The matrix \(A\) is \(\begin{pmatrix}2&1\\4&3\end{pmatrix}\).

(i) Find \((2A)^{-1}\).

(ii) Hence solve the simultaneous equations

\(2y+4x+5=0,\)

\(6y+8x+9=0.\)