ilovepdf_merged (6).pdf). Content covers sections 20.1–20.x.A matrix is a rectangular array of numbers arranged in m rows and n columns. This lesson covers types of matrices, equality, addition, scalar multiplication, multiplication, transpose, and elementary ideas used later for determinants and inverses.
Addition: only for same order; add corresponding entries. Scalar multiple: kA multiplies every entry by k.
Multiplication: AB defined when columns of A = rows of B. Entry (i,j) of AB is the dot product of row i of A with column j of B. In general AB ≠ BA.
Aᵀ is obtained by writing rows of A as columns. (AB)ᵀ = BᵀAᵀ. (Aᵀ)ᵀ = A.
Most exam-important points from this chapter:
Start every answer with the key definition or standard form from Matrices.
Copy the formula, then substitute values; never jump to the number alone.
Check domain, quadrant, non-zero denominators, and applicability of the theorem.
Memorise and apply: Find the minors of the elements of the 3rd row of matrix
Memorise and apply: Find the cofactors of elements of the 2nd row of matrix
English questions from NIOS Mathematics (311) public / sample papers (Sample 2024, Apr 2024, Oct 2024, Apr 2025; board-style where scanned papers had no extractable text). Mapped exclusively to this chapter — no cross-chapter duplicates. Use Model Answer for the marking key and Explanation for working.
5 question(s) · Sources: Board-style multiplication, Board-style transpose property, Sample QP 2024
PYQ1. If A = [[1, 2], [3, 4]], then A + Aᵀ equals:
Model Answer
1) [[2, 5], [5, 8]]
Explanation
1) Aᵀ=[[1,3],[2,4]]. A+Aᵀ=[[2,5],[5,8]].
PYQ2. If [[p+q, 2], [5, q]] = [[6, 2], [5, 2]], find p.
Model Answer
1) p = 4
Explanation
1) Equating: q=2 and p+q=6 ⇒ p+2=6 ⇒ p=4.
PYQ3. If A = [[1, 0], [0, 0]] and B = [[0, 0], [0, 1]], find AB.
Model Answer
1) [[0, 0], [0, 0]] (zero matrix)
Explanation
1) Row-column products: first row of A times columns of B give zeros
2) second row of A is zero. Hence AB = O.
PYQ4. State (AB)ᵀ in terms of Aᵀ and Bᵀ (when AB is defined).
Model Answer
1) (AB)ᵀ = Bᵀ Aᵀ
Explanation
1) Transpose of a product reverses the order of the factors.
PYQ5. If A=[[1,2],[0,1]] and B=[[3,0],[1,4]], compute AB.
Model Answer
1) [[5, 8], [1, 4]]
Explanation
1) AB₁₁=1·3+2·1=5; AB₁₂=1·0+2·4=8; AB₂₁=0·3+1·1=1; AB₂₂=0·0+1·4=4.
10 English exam-style problems for this chapter (NIOS Mathematics 311). Each question states the formula, gives full step-by-step working with the final answer, plus formal/easy explanations. English only.
If A = [[1,2,3],[0,−1,4]], write the order of A and the entry a₂₃.
1) A has 2 rows and 3 columns ⇒ order 2×3.
2) a₂₃ is row 2, column 3 = 4.
Answer: Order 2×3; a₂₃ = 4
Order is m×n; aᵢⱼ is the element in row i, column j.
Two rows, three columns; bottom-right-ish entry in row2 col3 is 4.
Indexing starts at 1 in standard school notation.
Do not confuse a₂₃ with a₃₂ (which does not exist here).
If A=[[1,2],[3,4]] and B=[[0,−1],[5,2]], find A+B.
1) A+B = [[1+0, 2+(−1)],[3+5, 4+2]] = [[1,1],[8,6]].
Answer: [[1, 1], [8, 6]]
Matrix addition is entrywise for equal order matrices.
Add numbers in the same positions.
A−B subtracts entrywise.
Orders must match.
Compute AB if A=[[1,2],[0,1]] and B=[[3,0],[1,4]].
1) AB₁₁ = 1·3+2·1 = 5; AB₁₂ = 1·0+2·4 = 8.
2) AB₂₁ = 0·3+1·1 = 1; AB₂₂ = 0·0+1·4 = 4.
3) AB = [[5,8],[1,4]].
Answer: [[5, 8], [1, 4]]
Row-by-column multiplication yields [[5,8],[1,4]].
Dot each row of A with each column of B.
AB and BA need not be equal; check BA separately if asked.
Inner dimensions must agree (here both 2×2).
Find the transpose of A=[[1,2,3],[4,5,6]].
1) Rows become columns: Aᵀ = [[1,4],[2,5],[3,6]].
2) Order becomes 3×2.
Answer: [[1,4],[2,5],[3,6]] (order 3×2)
Transpose interchanges rows and columns.
Write the first row downward as the first column, etc.
(Aᵀ)ᵀ=A; (AB)ᵀ=BᵀAᵀ.
Order flips from m×n to n×m.
If A=[[2,−1],[0,3]], find 3A.
1) 3A = [[6,−3],[0,9]].
Answer: [[6, −3], [0, 9]]
Scalar multiplication multiplies every entry by the scalar.
Triple each number in the matrix.
Distributes over addition: k(A+B)=kA+kB.
Sign of −1 becomes −3.
Write the 2×2 identity matrix I₂ and compute I₂[[2,1],[0,3]].
1) I₂ = [[1,0],[0,1]].
2) I₂A = A = [[2,1],[0,3]].
Answer: I₂=[[1,0],[0,1]]; product = [[2,1],[0,3]]
The identity acts as the multiplicative neutral element for compatible matrices.
Ones on the diagonal, zeros elsewhere; multiplying by I leaves A unchanged.
Only square identity matrices are used this way.
Do not write all ones.
If A = [[1, 0], [2, 3]] and B = [[2, 1], [0, 4]], find AB.
1) Row1·col1: 1·2+0·0=2; row1·col2: 1·1+0·4=1.
2) Row2·col1: 2·2+3·0=4; row2·col2: 2·1+3·4=14.
3) AB = [[2, 1], [4, 14]].
Answer: [[2, 1], [4, 14]]
Matrix product by row–column multiplication.
Each entry is a dot product.
Order 2×2 times 2×2 gives 2×2.
Check each algebraic step carefully.
Find the transpose of [[1, 2, 3], [4, 5, 6]].
1) Rows become columns: [[1, 4], [2, 5], [3, 6]].
Answer: [[1, 4], [2, 5], [3, 6]]
Transpose flips rows and columns.
3 columns become 3 rows.
Order changes from 2×3 to 3×2.
Check each algebraic step carefully.
If A = [[2, −1], [0, 3]], find −2A.
1) −2A = [[−4, 2], [0, −6]].
Answer: [[−4, 2], [0, −6]]
Scalar multiplication acts entrywise.
Multiply each number by −2.
Signs change carefully.
Check each algebraic step carefully.
Can we form AB if A is 2×3 and B is 2×3? Explain.
1) A has 3 columns; B has 2 rows.
2) 3 ≠ 2 ⇒ AB is not defined.
Answer: No — AB not defined
Inner dimensions must match for a product.
Need B to have 3 rows.
BA would need columns(B)=rows(A) ⇒ 3=2, also false.
Check each algebraic step carefully.