Matrices and determinants

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.

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? \[ \left (\array{ 1& 0 & 0\cr 0 & 1 & 1 \cr a& b & c} \right ), \left (\array{ d& e & f\cr -1 & 1 & -1 \cr 1& 0 & 1} \right ) \]
\(a=-1\), \(b=0\), \(c=1\), \(d=1\), \(e=0\), \(f=0\)
\(a=1\), \(b=0\), \(c=1\), \(d=-1\), \(e=0\), \(f=1\)
\(a=-1\), \(b=0\), \(c=-1\), \(d=1\), \(e=0\), \(f=0\)
\(a=1\), \(b=0\), \(c=-1\), \(d=-1\), \(e=1\), \(f=0\)

2000017107

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

2000017108

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

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?
\(a=2\), \(b=1\), \(c=0\)
\(a=1\), \(b=2\), \(c=1\)
\(a=2\), \(b=1\), \(c=3\)
\(a=1\), \(b=1\), \(c=0\)

2000017210

Level: 
B
In the following matrix, what is the position of the entry with the largest value? \[ \left (\array{ 1+2& \sin \frac{\pi}4& \mathrm{tg}\,\frac{3\pi}4\cr 3+\cos \frac{\pi}2 & 2^{\sqrt3} & 5+\mathrm{tg} (-\pi)\cr \sqrt{23}& 3-4 & \sin 2\pi -2 } \right ) \]
It lays above the main diagonal.
It lays on the main diagonal.
It lays under the main diagonal.
It lays on the counterdiagonal.

2000017410

Level: 
B
Find the matrix \(M\) for which the following equality is true: \[ 3 \cdot \left (\array{ 4& -1\cr 2 & 5 \cr } \right ) - 2\cdot M = \left (\array{ 14& -7 \cr 4 & 7 \cr } \right ) \]
\( \left (\array{ -1& 2\cr 1 & 4 \cr } \right ) \)
\( \left (\array{ 1& 2\cr 1 & 4 \cr } \right ) \)
\( \left (\array{ -1& 2\cr -1 & 4 \cr } \right ) \)
\( \left (\array{ -1& -2\cr 1 & 4 \cr } \right ) \)

2000017411

Level: 
B
For which matrix \(M\) the following equality holds? \[ 2 \cdot \left (\array{ 5& -4\cr 7 & -3 \cr } \right ) - 4\cdot M = \left (\array{ 2& -20 \cr 18 & -22 \cr } \right ) \]
\( \left (\array{ 2& 3\cr -1 & 4 \cr } \right ) \)
\( \left (\array{ 2& 3\cr 1 & 4 \cr } \right ) \)
\( \left (\array{ 2& -3\cr -1 & 4 \cr } \right ) \)
\( \left (\array{ -2& 3\cr -1 & 4 \cr } \right ) \)