Matrices and determinants

2000017406

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

2000017407

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

2000017408

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

2000018301

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

2000018302

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

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 \(m=-\frac12\)

2000018305

Level: 
A
Consider three matrices \[ A = \left (\array{ 3 &4\cr 1 & 2\cr } \right ),~ B = \left (\array{ 1 &1\cr 0&1\cr } \right ),~ C = \left (\array{ 1 &0\cr 1&1\cr } \right ). \] Let \(E\) denote the identity matrix of order \(2\). Then, find \(X\), which is the solution to the following equation. \[ C \cdot (A+X)\cdot B=E\]
\( X = \left (\array{ -2 &-5\cr -2& 0\cr } \right ) \)
\( X = \left (\array{ -2 &-5\cr 2& 0\cr } \right ) \)
\( X = \left (\array{ -2 &5\cr -2& 0\cr } \right ) \)
\( X = \left (\array{ -2 &5\cr 2& 0\cr } \right ) \)

2000019301

Level: 
A
Three ice cream stands of ICE company reported their July sales of four ice cream flavors in number of portions sold. All data can be seen in the table below. \[ \begin{array}{|c|c|c|c|c|} \hline &\text{vanilla} & \text{chocolate} & \text{nut} & \text{strawberry} \\\hline \text{Stand 1}& 720 & 800 & 1\,200&360 \\\hline \text{Stand 2} & 550 & 434 & 900 & 300 \\\hline \text{Stand 3} &610 &300 & 200 & 750 \\\hline \end{array}\] The data provided by the stands for August sales are reported in the matrix \(A\). \[ A= \left (\array{ 650& 470 & 890 & 410\cr 500& 505 & 890 & 300\cr 380& 520 & 350 & 800\cr } \right ) \] If the July sales are rewritten to the matrix \(J\), by what matrix the sales of ice cream for both summer months are described?
matrix \(J+A\)
matrix \(J-A\)
matrix \(J \cdot A\)
matrix \(2J+2A\)