Matrices and determinants

2000017206

Level: 
B
For what values of \(a\), \(b\) and \(c\) is the matrix \[ \left (\array{ a-b+c& 0 & a-2b\cr a+c-2b & 2^{a-2b} & 0\cr 2b-a & 0 & c-a+3b } \right ) \] an identity matrix?
\(a=2\), \(b=1\), \(c=0\)
\(a=1\), \(b=2\), \(c=1\)
\(a=2\), \(b=1\), \(c=3\)
\(a=1\), \(b=1\), \(c=0\)

2000017205

Level: 
A
Which of the given matrices is the matrix \((m_{i,j})\), where \(i=1, \dots, 3\) and \(j=1,\dots,3\) with \(m_{i,j}=i+j+1\)?
\( \left (\array{ 3& 4 & 5\cr 4 & 5 & 6\cr 5 & 6 & 7 } \right ) \)
\( \left (\array{ 2& 3 & 4\cr 3 & 4 & 5\cr 4 & 5 & 6 } \right ) \)
\( \left (\array{ 3& 4 & 5\cr 3 & 4 & 5\cr 4 & 5 & 6 } \right ) \)
\( \left (\array{ 2& 3 & 4\cr 2 & 3 & 4\cr 4 & 5 & 6 } \right ) \)

2000017204

Level: 
A
Which of the given matrices is the matrix \((m_{i,j})\), where \(i=1, \dots, 3\) and \(j=1,~2\)?
\( \left (\array{ 8& 7\cr 6 & 5\cr 4 & 3\cr } \right ) \)
\( \left (\array{ 8& 7 & 6\cr 5 & 4 & 3\cr } \right ) \)
\( \left (\array{ 8& 7 & 6\cr 5 & 4 & 3\cr 2 & 1 & 0 } \right ) \)
\( \left (\array{ 8& 7 \cr 6 & 5 \cr } \right ) \)

2000017203

Level: 
A
Which of the given matrices have the same entry at the position \( (1,2)\)? \[ K=\left (\array{ 1& \sqrt2 & 3 & \sqrt5\cr \sqrt3& 2 & 1 & 5\cr 4& 1 & 1& 0\cr } \right ), \quad L=\left (\array{ 1& \sqrt2 & 3\cr \sqrt3 & 2 & 1\cr 4 & 1& 1\cr \sqrt5 & 5& 0 } \right ), \] \[ M=\left (\array{ \sqrt3& 2 & 1\cr \sqrt3 & 4 & 0 } \right ), \quad N=\left (\array{ 1& \sqrt3 & 4\cr \sqrt3 & 2 & 1 \cr 3 & 1 & 1 \cr \sqrt5 & 5 & 0 } \right ) \]
\(K\) and \(L\)
\(K\), \(L\) and \(N\)
\(K\), \(L\), \(M\), and \(N\)
\(L\) and \(N\)

2000017202

Level: 
A
Which of the given matrices \(K\), \(L\), \(M\), and \(N\) have the same main diagonal? \[ K=\left (\array{ 1& 7 & 8\cr 4 & 2 & 9 \cr 5 & 6 & 3 } \right ),\quad L=\left (\array{ 3& 9 & 8\cr 4 & 2 & 7 \cr 5 & 6 & 1 } \right ), \] \[ M=\left (\array{ 1& 7 & 9\cr 5 & 2 & 8 \cr 4 & 6 & 3 } \right ), \quad N=\left (\array{ 2& 7 & 8\cr 4 & 2 & 6 \cr 5 & 7 & 5 } \right ) \]
\(K\) and \(M\)
\(K\), \(L\) and \(M\)
\(K\), \(L\) and \(N\)
The main diagonals of any two matrices differ.

2000017201

Level: 
A
Which of the given matrices is of order \(3\) and has an entry of \(2\) at the position \((3,2)\)?
\[ \left (\array{ 1& 2 & 3\cr 4 & 3 & 4 \cr 3 & 2 & 1 } \right ) \]
\[ \left (\array{ 1& 2 & 3\cr 4 & 3 & 2 \cr 3 & 4 & 1 } \right ) \]
\[ \left (\array{ 1& 2 & 3\cr 4 & 3 & 2 \cr 3 & 2 & 2 \cr 1 & 2 & 3} \right ) \]
\[ \left (\array{ 1& 2 \cr 3 & 4 \cr 3 & 2 } \right ) \]

2000017108

Level: 
B
For what values of the real parameters \(a\) and \(b\) will the matrices given below be inverse to each other? \[ \left (\array{ a& 7 \cr 3 & 1 \cr} \right ), ~ \left (\array{ -\frac1{16}& \frac7{16} \cr b & -\frac5{16} \cr} \right ) \]
\(a=5\), \(b=\frac3{16}\)
\(a=-5\), \(b=\frac3{16}\)
\(a=5\), \(b=-\frac3{16}\)
\(a=-5\), \(b=-\frac3{16}\)

2000017107

Level: 
B
Compute the inverse matrix to the matrix: \[ \left (\array{ \frac17& -\frac3{14}\cr \frac27 & \frac1{14}} \right ) \]
\[ \left (\array{ 1& 3\cr -4& 2}\right ) \]
\[ \left (\array{ 1& 3\cr 4& -2}\right ) \]
\[ \left (\array{ 1& 3\cr 2& -4}\right ) \]
\[ \left (\array{ 2& -3\cr 4& 1}\right ) \]

2000017106

Level: 
B
For what values of the real parameters \(a\), \(b\), \(c\), \(d\), \(e\) and \(f\) will the matrices given below be inverse to each other? \[ \left (\array{ 1& 0 & 0\cr 0 & 1 & 1 \cr a& b & c} \right ), \left (\array{ d& e & f\cr -1 & 1 & -1 \cr 1& 0 & 1} \right ) \]
\(a=-1\), \(b=0\), \(c=1\), \(d=1\), \(e=0\), \(f=0\)
\(a=1\), \(b=0\), \(c=1\), \(d=-1\), \(e=0\), \(f=1\)
\(a=-1\), \(b=0\), \(c=-1\), \(d=1\), \(e=0\), \(f=0\)
\(a=1\), \(b=0\), \(c=-1\), \(d=-1\), \(e=1\), \(f=0\)

2000017105

Level: 
B
Let \(x\) be a real number. Determine the inverse matrix to the matrix: \[ \left (\array{ \cos x& -\sin x \cr \sin x & \cos x \cr} \right ) \]
\[ \left (\array{ \cos x& \sin x \cr -\sin x & \cos x \cr} \right ) \]
\[ \left (\array{ 1& 0 \cr 0 & 1 \cr} \right ) \]
\[ \left (\array{ \frac{\cos x}{\sin 2x}& \frac{\sin x}{\sin 2x}\cr -\frac{\sin x}{\sin 2x}& \frac{\cos x}{\sin 2x} \cr} \right ) \]
The inverse matrix does not exist.