Level 1 — RecognitionMatrices & Determinants — Introduction

Matrices & Determinants — Introduction

30 marksprintable — key stays hidden on paper

Level 1 — Recognition

Time: 20 minutes Total Marks: 30

Section A — Multiple Choice (1 mark each)

Choose the single best answer.

Q1. The order of the matrix $\begin{pmatrix} 1 & 4 & 7 \\ 2 & 5 & 8 \end{pmatrix}$ is: (a) $3\times 2$ (b) $2\times 3$ (c) $6\times 1$ (d) $2\times 2$

Q2. A square matrix $A$ is called symmetric if: (a) $A^{T}=-A$ (b) $A^{T}=A$ (c) $A^{2}=A$ (d) $A^{T}=A^{-1}$

Q3. Two matrices can be added only if: (a) they are both square (b) they have the same number of columns only (c) they have the same order (d) the number of columns of the first equals the rows of the second

Q4. The product $AB$ is defined when $A$ is $m\times n$ and $B$ is $p\times q$ only if: (a) $m=p$ (b) $n=q$ (c) $n=p$ (d) $m=q$

Q5. The determinant of $\begin{pmatrix} 3 & 2 \\ 1 & 4 \end{pmatrix}$ is: (a) $10$ (b) $14$ (c) $-10$ (d) $12$

Q6. The identity matrix of order 2 is: (a) $\begin{pmatrix}0&0\\0&0\end{pmatrix}$ (b) $\begin{pmatrix}1&1\\1&1\end{pmatrix}$ (c) $\begin{pmatrix}1&0\\0&1\end{pmatrix}$ (d) $\begin{pmatrix}1&0\\1&0\end{pmatrix}$

Q7. For a $2\times2$ matrix $A=\begin{pmatrix}a&b\\c&d\end{pmatrix}$ , $A^{-1}$ exists provided: (a) $ad-bc=0$ (b) $ad-bc\neq0$ (c) $a=d$ (d) $b=c$

Q8. A skew-symmetric matrix must have diagonal entries all equal to: (a) $1$ (b) each other (c) $0$ (d) $-1$

Q9. If $k$ is a scalar, then $\det(kA)$ for a $2\times2$ matrix $A$ equals: (a) $k\det A$ (b) $k^{2}\det A$ (c) $\det A$ (d) $2k\det A$

Q10. The transpose of $\begin{pmatrix}1&2\\3&4\\5&6\end{pmatrix}$ has order: (a) $3\times2$ (b) $2\times3$ (c) $3\times3$ (d) $2\times2$

Section B — Matching (1 mark each, 5 marks)

Q11. Match each matrix type in Column X to its defining property in Column Y.

Column X	Column Y
(i) Row matrix	(P) all elements zero
(ii) Diagonal matrix	(Q) only one row
(iii) Zero matrix	(R) $A^T = -A$
(iv) Identity matrix	(S) non-diagonal elements zero
(v) Skew-symmetric	(T) diagonal matrix with all diagonal entries 1

Section C — True/False with justification (2 marks each: 1 verdict + 1 reason)

Q12. Matrix multiplication is commutative, i.e. $AB=BA$ always.

Q13. If $\det A = 0$ then the matrix $A$ has no inverse.

Q14. $(A^{T})^{T}=A$ for every matrix $A$ .

Q15. A system $AX=B$ can be solved by $X=A^{-1}B$ even when $\det A=0$ .

Q16. In Cramer's rule for a $2\times2$ system, the solution is $x=\dfrac{D_x}{D}$ where $D$ is the determinant of the coefficient matrix.

Answer keyMark scheme & solutions

Section A (1 mark each)

Q1 — (b) $2\times3$ . Order = rows × columns = 2 rows, 3 columns. ✔(1)

Q2 — (b) $A^T=A$ . Definition of symmetric matrix. ✔(1)

Q3 — (c) same order. Addition is entry-wise, so both dimensions must match. ✔(1)

Q4 — (c) $n=p$ . Inner dimensions must agree; result is $m\times q$ . ✔(1)

Q5 — (a) $10$ . $\det = (3)(4)-(2)(1)=12-2=10$ . ✔(1)

Q6 — (c) $\begin{pmatrix}1&0\\0&1\end{pmatrix}$ . 1's on diagonal, 0 elsewhere. ✔(1)

Q7 — (b) $ad-bc\neq0$ . Inverse exists iff determinant nonzero. ✔(1)

Q8 — (c) $0$ . $A^T=-A$ forces $a_{ii}=-a_{ii}\Rightarrow a_{ii}=0$ . ✔(1)

Q9 — (b) $k^2\det A$ . Each of 2 rows scales by $k$ ; $\det(kA)=k^n\det A$ with $n=2$ . ✔(1)

Q10 — (b) $2\times3$ . Transpose swaps order $3\times2\to2\times3$ . ✔(1)

Section B (1 mark each)

Q11: (i)→Q, (ii)→S, (iii)→P, (iv)→T, (v)→R. ✔(5) Reasoning: row matrix = one row; diagonal = off-diagonal zero; zero = all zero; identity = diagonal of 1's; skew-symmetric = $A^T=-A$ .

Section C (2 marks each: verdict 1 + reason 1)

Q12 — FALSE. ✔(1) Matrix multiplication is generally non-commutative; order matters, and $AB$ may even have different dimensions than $BA$ . Reason ✔(1)

Q13 — TRUE. ✔(1) The inverse formula divides by $\det A$ ; if $\det A=0$ the matrix is singular and non-invertible. Reason ✔(1)

Q14 — TRUE. ✔(1) Transposing twice restores original positions of all elements. Reason ✔(1)

Q15 — FALSE. ✔(1) $A^{-1}$ does not exist when $\det A=0$ , so the method fails (no unique solution). Reason ✔(1)

Q16 — TRUE. ✔(1) Cramer's rule: $x=D_x/D$ , $y=D_y/D$ , where $D$ is the coefficient-matrix determinant ( $D\neq0$ ). Reason ✔(1)

[
  {"claim":"det of [[3,2],[1,4]] is 10","code":"M=Matrix([[3,2],[1,4]]); result=(M.det()==10)"},
  {"claim":"det(kA)=k**2 det A for 2x2","code":"k=symbols('k'); A=Matrix([[symbols('a'),symbols('b')],[symbols('c'),symbols('d')]]); result=simplify((k*A).det()-k**2*A.det())==0"},
  {"claim":"transpose of 3x2 has shape 2x3","code":"M=Matrix([[1,2],[3,4],[5,6]]); result=(M.T.shape==(2,3))"},
  {"claim":"double transpose returns original","code":"M=Matrix([[1,2,3],[4,5,6]]); result=(M.T.T==M)"}
]