A

2000017407

Level: 
A
Find \(A^2-B^2\), if: \[ A=\left (\array{ 2& 1\cr 3 & 0 \cr } \right ) , B=\left (\array{ 1& 4 \cr 2 & -1 \cr } \right ) \]
\( \left (\array{ -2& 2 \cr 6 & -6\cr } \right ) \)
\( \left (\array{ -2& 2 \cr 6 & 6\cr } \right ) \)
\( \left (\array{ -2& 2 \cr 6 & -4\cr } \right ) \)
\( \left (\array{ 2& 2 \cr 6 & 6\cr } \right ) \)

2000017406

Level: 
A
For which numbers \(x\) and \(y\) is the following equality true? \[ \left (\array{ 2& 3\cr 1 & -5 \cr } \right ) \cdot \left (\array{ x\cr y \cr } \right ) = \left (\array{ 1.5& x \cr y & 5 \cr } \right ) \cdot \left (\array{ -2\cr -1 \cr } \right ) \]
\(x=-2\), \(y=1\)
\(x=2\), \(y=1\)
\(x=-5\), \(y=4\)
\(x=-2\), \(y=-1\)

2000017404

Level: 
A
Consider the matrices: \[ A=\left (\array{ 1& -1 & 2\cr 3 & 4 & 1 \cr 0 & 1 & 0} \right ), \ B=\left (\array{ 2& 3 & -1\cr -2 & 1 & 1 \cr 1 & 4 & -1 } \right ). \] Find the transpose of the matrix \(A+B\).
\( \left (\array{ 3& 1 & 1\cr 2 & 5& 5 \cr 1 & 2& -1} \right ) \)
\( \left (\array{ 3& 2 & 1\cr 1& 5 & 2\cr 1 & 5& -1 } \right ) \)
\( \left (\array{ 3& 1 & 1\cr 2 & 5&5 \cr 1 & 2& 1 } \right ) \)
\( \left (\array{ 3& 2 & 1\cr 2 &5& 2 \cr 1&2& -1 } \right ) \)

2000017403

Level: 
A
For which real numbers \(a\), \(b\), \(c\), \(d\) is the following equality true? \[ 2 \cdot \left (\array{ a& c\cr b & d \cr } \right ) - \left (\array{ 2& 1 \cr 3 & 5 \cr } \right ) = \left (\array{ 4& 7\cr -5 & -5\cr } \right ) \]
\(a=3\), \(b=-1\), \(c=4\), \(d=0\)
\(a=1\), \(b=1\), \(c=4\), \(d=1\)
\(a=3\), \(b=-4\), \(c=4\), \(d=0\)
\(a=3\), \(b=-1\), \(c=4\), \(d=1\)

2000017402

Level: 
A
Find the product: \[ \left (\array{ 1.2& 1 & 0\cr 0.3 & 0.1 & 1 \cr 2 & 0 & 4} \right ) \cdot \left (\array{ 1& 1 & 0\cr 0 & 1 & 1 \cr 1 & 1 & 1 } \right ) \]
\( \left (\array{ 1.2& 2.2 & 1\cr 1.3 & 1.4& 1.1 \cr 6 & 6& 4 } \right ) \)
\( \left (\array{ 1.2& 1 & 0\cr 0& 0.1 & 1\cr 2 & 0& 4 } \right ) \)
\( \left (\array{ 3.2& 3.2 & 1\cr 1.4 & 1.4& 1.4 \cr 6 & 7& 4 } \right ) \)
\( \left (\array{ 1.2& 2.2 & 1\cr 1.1 & 1.4& 0 \cr 2 & 4& 6 } \right ) \)

2000017401

Level: 
A
Find the product of the following matrices: \[ A=\left (\array{ \frac12 & \frac34\cr -1 & \frac32 } \right ),~ B=\left (\array{ \frac32 & -\frac14 \cr 0& \frac12 } \right ) \]
\( AB=\left (\array{ \frac34 & \frac14 \cr -\frac32& 1 } \right ) \)
\( AB=\left (\array{ \frac34 & -\frac3{16} \cr 0 & \frac34} \right ) \)
\( AB=\left (\array{ \frac34 & \frac12 \cr -\frac32 & 1 } \right ) \)
\( AB=\left (\array{ 2 & \frac12 \cr -1 & 2} \right ) \)

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 \(m_{i,j}=i+j+1\)?
\( \left (\array{ 3& 4 & 5\cr 4 & 5 & 6\cr 5 & 6 & 7 } \right ) \)
\( \left (\array{ 2& 3 & 4\cr 3 & 4 & 5\cr 4 & 5 & 6 } \right ) \)
\( \left (\array{ 3& 4 & 5\cr 3 & 4 & 5\cr 4 & 5 & 6 } \right ) \)
\( \left (\array{ 2& 3 & 4\cr 2 & 3 & 4\cr 4 & 5 & 6 } \right ) \)