Matrices and determinants

2000018303

Level: 
A
Let \(E\) denote an identity matrix of order \(2\) and let matrix \[ M = \left (\array{ m &0\cr 0 & 2\cr } \right ) . \] Find all the values of \(m\) so that the equality below holds. \[ M^2-\frac52M+E=0 \]
\(m=2\) or \(m=\frac12\)
\(m=\frac12\)
\(m=2\)
\(m=2\) or \(m=-\frac12\)

2000018302

Level: 
A
Find the matrix \(M\) so that the equality given below is true. \[ 2 \cdot \left (\array{ -1&4\cr 3&-5\cr } \right ) - M = \left (\array{ -3 &6\cr 9 & -14\cr } \right ) \]
\[ M=\left (\array{ 1 &2\cr -3 & 4\cr } \right ) \]
\[ M=\left (\array{ -1 &2\cr -3 & 4\cr } \right ) \]
\[ M=\left (\array{ -1 &-2\cr 3 & -4\cr } \right ) \]
\[ M=\left (\array{ 1 &2\cr 3 & -4\cr } \right ) \]

2000018301

Level: 
A
Find the matrix \(B\), the solution to the equation given below. \[ \left (\array{ 3&-1 &5\cr 1 &0&3 } \right ) + B = \left (\array{ 5 & 0 & 4 \cr 3 & 2 & 1\cr } \right ) \]
\[ B= \left (\array{ 2 & 1 & -1\cr 2 & 2 & -2 } \right ) \]
\[ B= \left (\array{ 2 & -1 & -1\cr 2 & 2 & -2 } \right ) \]
\[ B= \left (\array{ 2 & 1 & -1\cr 2 & -2 & -2 } \right ) \]
\[ B= \left (\array{ 2 & 1 & -1\cr 2 & 2 & 2 } \right ) \]

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.