Matrices and determinants

2000017606

Level: 
C
Determine the set of all real numbers \(b\) for which the determinant of the following matrix equals to \(5\). \[ \left (\array{ 4 & b & -1\cr 3 &0& 2\cr b & 0 & -1\cr } \right ) \]
\( \left\{1;-\frac52\right\}\)
\( \left\{-\frac52\right\}\)
\( \left\{1\right\}\)
\( \emptyset\)

2000017604

Level: 
C
Which of the given matrices \(A\), \(B\), \(C\) and \(D\) has a different determinant than the others? \[ A=\left (\array{ 6 & 11 \cr 2 & 2\cr } \right ) \] \[ B=\left (\array{ 1 & 3 \cr 5 & 2\cr } \right ) \] \[ C=\left (\array{ 5 & -2 \cr 10 & -6\cr } \right ) \] \[ D=\left (\array{ 10 & 0 \cr -7 & -1\cr } \right ) \]
\( B\)
\( D\)
All given matrices have the same determinant.
Each matrix has a different determinant.

2000017413

Level: 
B
Consider the matrices \[ A=\left (\array{ 1& 0 & 0\cr -1 & 1 & 1 \cr 2& 1 & 0} \right ),~ B=\left (\array{ -1& 1 & 0\cr 0 & 1 & 1 \cr 1 &1& -1 } \right ). \] For which matrix \(X\) holds \(A \cdot B - X= B^2\)?
\( \left (\array{ -2& 1 & -1\cr 1 & -1& 0\cr 0 & 2& -1 } \right ) \)
\( \left (\array{ -2& 1 & -1\cr 1 & 1& 0\cr 0 & 2& -1 } \right ) \)
\( \left (\array{ -2& 1 & 1\cr 1 & -1& 0\cr 0 & 2& -1 } \right ) \)
\( \left (\array{ 2& 1 & -1\cr 1 & -1& 0\cr 0 & 2& -1 } \right ) \)

2000017408

Level: 
A
Find \(\frac{K \cdot (-L)}2\), if: \[ K=\left (\array{ 3& 0 & 1\cr 2 & 3 & 4 \cr 1& -1 & 1} \right ),~ L=\left (\array{ 2& 3 & 0\cr 1 & 1 & -1 \cr 2 &0& 1 } \right ) \]
\( \left (\array{ -4& -4.5 & -0.5\cr -7.5 & -4.5& -0.5\cr -1.5 & -1& -1 } \right ) \)
\( \left (\array{ -4& 4.5 & 0.5\cr -7.5 & -4.5& -0.5\cr -1.5 & -1& -1 } \right ) \)
\( \left (\array{ -4& -4.5 & -0.5\cr 7.5 & -4.5& -0.5\cr -1.5 & -1& -1 } \right ) \)
\( \left (\array{ -4& -4.5 & -0.5\cr -7.5 & 4.5& -0.5\cr -1.5 & -1& 1 } \right ) \)

2000017412

Level: 
B
For which matrices \(X\) and \(Y\) are both following equations true? \[ 3\cdot X - Y= \left (\array{ 2& -2 & -5\cr 8 & 3 & 2\cr } \right ) \] \[ X + 2 \cdot Y= \left (\array{ 3& 4& -4\cr 5 & 1 & 3\cr } \right ) \]
\(X= \left (\array{ 1& 0& -2\cr 3 & 1 & 1\cr } \right ) \), \(Y= \left (\array{ 1& 2& -1\cr 1 & 0 & 1\cr } \right ) \)
\(X= \left (\array{ 1& 0& 2\cr 3 & 1 & 1\cr } \right ) \), \(Y= \left (\array{ 1& 2& -1\cr 1 & 0 & 1\cr } \right ) \)
\(X= \left (\array{ 1& 0& -2\cr 3 & 1 & 1\cr } \right ) \), \(Y= \left (\array{ 1& 2& 1\cr 1 & 0 & 1\cr } \right ) \)
\(X= \left (\array{ 1& 0& -2\cr 3 & -1 & 1\cr } \right ) \), \(Y= \left (\array{ 1& 2& -1\cr -1 & 0 & 1\cr } \right ) \)