Matrices and determinants

2000017412

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

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 \(A \cdot B - X= B^2\)?
\( \left (\array{ -2& 1 & -1\cr 1 & -1& 0\cr 0 & 2& -1 } \right ) \)
\( \left (\array{ -2& 1 & -1\cr 1 & 1& 0\cr 0 & 2& -1 } \right ) \)
\( \left (\array{ -2& 1 & 1\cr 1 & -1& 0\cr 0 & 2& -1 } \right ) \)
\( \left (\array{ 2& 1 & -1\cr 1 & -1& 0\cr 0 & 2& -1 } \right ) \)

2000018304

Level: 
B
Which matrices X and Y make valid both equalities given below? \[ 2X+Y = \left (\array{ 1 &4\cr 2 & 0\cr } \right ) \] \[ X-Y = \left (\array{ 1 &-1\cr 1 & 0\cr } \right ) \]
\( X = \left (\array{ \frac23 &1\cr 1 & 0\cr } \right ) \) and \( Y = \left (\array{ -\frac13 &2\cr 0& 0\cr } \right ) \)
\( X = \left (\array{ \frac23 &1\cr 1 & 0\cr } \right ) \) and \( Y = \left (\array{ -\frac13 &4\cr 0& 0\cr } \right ) \)
\( X = \left (\array{ \frac23 &1\cr 1 & 0\cr } \right ) \) and \( Y = \left (\array{ \frac13 &2\cr 0& 0\cr } \right ) \)
\( X = \left (\array{ \frac23 &1\cr 1 & 1\cr } \right ) \) and \( Y = \left (\array{ -\frac13 &4\cr 0& 0\cr } \right ) \)

2000018906

Level: 
B
Specify how the rank of the matrix \(A\) changes depending on the value of \(t\), where \[ A=\left (\array{ 3& -2& 1&-4\cr -6& 4& -2&8\cr 0& t& 0&t} \right ). \]
If \(t=0\), the rank is \(1\), otherwise the rank is \(2\).
If \(t=0\), the rank is \(1\), otherwise the rank is \(3\).
If \(t=0\), the rank is \(2\), otherwise the rank is \(1\).
If \(t=2\), the rank is \(3\), otherwise the rank is \(1\).