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Level 2 — RecallMatrices & Determinants — Introduction

Matrices & Determinants — Introduction

40 marksprintable — key stays hidden on paper

Level 2 — Recall & Standard Problems

Time: 30 minutes Total Marks: 40

Q1. Define the order of a matrix. State the order of A=[2−15034]A = \begin{bmatrix} 2 & -1 & 5 \\ 0 & 3 & 4 \end{bmatrix} and name the element a23a_{23}. (3 marks)

Q2. Define a symmetric and a skew-symmetric matrix. Give one 2×22\times 2 example of each. (4 marks)

Q3. Given A=[1234]A = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix} and B=[0−152]B = \begin{bmatrix} 0 & -1 \\ 5 & 2 \end{bmatrix}, compute 2A−B2A - B. (3 marks)

Q4. For A=[1234]A = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix} and B=[2013]B = \begin{bmatrix} 2 & 0 \\ 1 & 3 \end{bmatrix}, compute ABAB and BABA. State whether matrix multiplication is commutative here. (5 marks)

Q5. Find the transpose of C=[147258]C = \begin{bmatrix} 1 & 4 & 7 \\ 2 & 5 & 8 \end{bmatrix} and verify that (CT)T=C(C^T)^T = C. (3 marks)

Q6. Evaluate the determinant ∣3524∣\begin{vmatrix} 3 & 5 \\ 2 & 4 \end{vmatrix}. (2 marks)

Q7. Evaluate the determinant of M=[123045106]M = \begin{bmatrix} 1 & 2 & 3 \\ 0 & 4 & 5 \\ 1 & 0 & 6 \end{bmatrix} by cofactor expansion along the first row. (5 marks)

Q8. Find the inverse of P=[4322]P = \begin{bmatrix} 4 & 3 \\ 2 & 2 \end{bmatrix}, if it exists. (4 marks)

Q9. Solve the following system using Cramer's rule: 2x+3y=8,x−y=−1.2x + 3y = 8, \qquad x - y = -1. (5 marks)

Q10. State any three properties of determinants. (3 marks)

End of Paper

Answer keyMark scheme & solutions

Q1. (3)

  • Order = number of rows × number of columns (rows first). (1)
  • AA has 2 rows, 3 columns → order 2×32\times 3. (1)
  • a23a_{23} = element in row 2, column 3 = 44. (1)

Q2. (4)

  • Symmetric: AT=AA^T = A (i.e. aij=ajia_{ij}=a_{ji}). (1) Example [1223]\begin{bmatrix}1&2\\2&3\end{bmatrix}. (1)
  • Skew-symmetric: AT=−AA^T = -A (i.e. aij=−ajia_{ij}=-a_{ji}, diagonal zero). (1) Example [02−20]\begin{bmatrix}0&2\\-2&0\end{bmatrix}. (1)

Q3. (3)

  • 2A=[2468]2A = \begin{bmatrix}2&4\\6&8\end{bmatrix}. (1)
  • Subtract element-wise: 2A−B=[2−04−(−1)6−58−2]=[2516]2A-B = \begin{bmatrix}2-0 & 4-(-1)\\6-5 & 8-2\end{bmatrix} = \begin{bmatrix}2&5\\1&6\end{bmatrix}. (2)

Q4. (5)

  • AB=[1⋅2+2⋅11⋅0+2⋅33⋅2+4⋅13⋅0+4⋅3]=[461012]AB = \begin{bmatrix}1\cdot2+2\cdot1 & 1\cdot0+2\cdot3\\3\cdot2+4\cdot1 & 3\cdot0+4\cdot3\end{bmatrix} = \begin{bmatrix}4&6\\10&12\end{bmatrix}. (2)
  • BA=[2⋅1+0⋅32⋅2+0⋅41⋅1+3⋅31⋅2+3⋅4]=[241014]BA = \begin{bmatrix}2\cdot1+0\cdot3 & 2\cdot2+0\cdot4\\1\cdot1+3\cdot3 & 1\cdot2+3\cdot4\end{bmatrix} = \begin{bmatrix}2&4\\10&14\end{bmatrix}. (2)
  • AB≠BAAB \neq BA → not commutative. (1)

Q5. (3)

  • CT=[124578]C^T = \begin{bmatrix}1&2\\4&5\\7&8\end{bmatrix} (rows↔columns). (2)
  • Transposing again returns rows to original: (CT)T=[147258]=C(C^T)^T = \begin{bmatrix}1&4&7\\2&5&8\end{bmatrix}=C. (1)

Q6. (2)

  • det⁡=3⋅4−5⋅2=12−10=2\det = 3\cdot4 - 5\cdot2 = 12-10 = 2. (2)

Q7. (5)

  • Expand along row 1: 1⋅∣4506∣−2⋅∣0516∣+3⋅∣0410∣1\cdot\begin{vmatrix}4&5\\0&6\end{vmatrix} - 2\cdot\begin{vmatrix}0&5\\1&6\end{vmatrix} + 3\cdot\begin{vmatrix}0&4\\1&0\end{vmatrix}. (2)
  • Minors: (4⋅6−5⋅0)=24(4\cdot6-5\cdot0)=24; (0⋅6−5⋅1)=−5(0\cdot6-5\cdot1)=-5; (0⋅0−4⋅1)=−4(0\cdot0-4\cdot1)=-4. (2)
  • =1(24)−2(−5)+3(−4)=24+10−12=22=1(24) -2(-5) +3(-4) = 24 +10 -12 = 22. (1)

Q8. (4)

  • det⁡P=4⋅2−3⋅2=8−6=2≠0\det P = 4\cdot2 - 3\cdot2 = 8-6 = 2 \neq 0 → inverse exists. (1)
  • Formula P−1=1det⁡[d−b−ca]P^{-1}=\frac{1}{\det}\begin{bmatrix}d&-b\\-c&a\end{bmatrix}. (1)
  • P−1=12[2−3−24]=[1−32−12]P^{-1}=\frac{1}{2}\begin{bmatrix}2&-3\\-2&4\end{bmatrix} = \begin{bmatrix}1&-\tfrac32\\-1&2\end{bmatrix}. (2)

Q9. (5)

  • D=∣231−1∣=−2−3=−5D=\begin{vmatrix}2&3\\1&-1\end{vmatrix}=-2-3=-5. (1)
  • Dx=∣83−1−1∣=−8−(−3)=−5D_x=\begin{vmatrix}8&3\\-1&-1\end{vmatrix}=-8-(-3)=-5. (1)
  • Dy=∣281−1∣=−2−8=−10D_y=\begin{vmatrix}2&8\\1&-1\end{vmatrix}=-2-8=-10. (1)
  • x=Dx/D=−5/−5=1x=D_x/D = -5/-5 = 1; y=Dy/D=−10/−5=2y=D_y/D = -10/-5 = 2. (2)

Q10. (3) Any three (1 each):

  • If two rows/columns are interchanged, determinant changes sign.
  • If a row/column is multiplied by kk, determinant multiplies by kk.
  • det⁡(AT)=det⁡(A)\det(A^T)=\det(A).
  • If any two rows/columns are identical, determinant =0=0.
  • det⁡(AB)=det⁡(A)det⁡(B)\det(AB)=\det(A)\det(B).
[
{"claim":"Q3: 2A-B equals [[2,5],[1,6]]","code":"A=Matrix([[1,2],[3,4]]);B=Matrix([[0,-1],[5,2]]);result=(2*A-B==Matrix([[2,5],[1,6]]))"},
{"claim":"Q4: AB and BA differ","code":"A=Matrix([[1,2],[3,4]]);B=Matrix([[2,0],[1,3]]);result=(A*B==Matrix([[4,6],[10,12]]) and B*A==Matrix([[2,4],[10,14]]) and A*B!=B*A)"},
{"claim":"Q7: det of M is 22","code":"M=Matrix([[1,2,3],[0,4,5],[1,0,6]]);result=(M.det()==22)"},
{"claim":"Q8: inverse of P","code":"P=Matrix([[4,3],[2,2]]);result=(P.inv()==Matrix([[1,Rational(-3,2)],[-1,2]]))"},
{"claim":"Q9: Cramer solution x=1,y=2","code":"result=(solve([2*x+3*y-8, x-y+1],[x,y])=={x:1,y:2})"}
]