Class 12 Maths · Chapter 3
Short answer:
Class 12 Maths Chapter 3 Matrices NCERT solutions — 2026-27 rationalised syllabus, 13 chapters me se 3rd. Is chapter me matrix ki definition, order, types (row, column, square, diagonal, scalar, identity, zero), addition/subtraction/scalar multiplication, matrix multiplication, transpose, symmetric aur skew-symmetric matrices, elementary operations ka brief idea, aur invertible matrices ka conceptual link (jo next chapter determinants me detail me aayega) cover hota hai. Neeche 18 exercise-style questions step-by-step solve kiye gaye hain — order check karna, addition/multiplication rules, transpose properties, symmetric-skew-symmetric proofs, aur matrix equations solve karna sab included hai.
Matrix ek rectangular array hai numbers ka, jo rows aur columns me arranged hote hain — aur ye array algebra ke rules follow karta hai. Class 11 me tumne numbers, sets, functions padhe; ab matrices ek naya "object" hai jispe apna addition, multiplication aur transpose define hota hai. Real life me matrices data organize karne ke liye use hote hain — jaise ek school ka marks table, ya ek company ka sales data city-wise aur month-wise. Is chapter ka sabse important concept hai matrix multiplication ka order-matching rule — jo agle kai chapters (determinants, linear equations) ki foundation banata hai.
Chapter 3 Summary — 5 Minute Revision
1. Matrix kya hai aur order
Matrix numbers (ya functions) ka rectangular arrangement hai, rows aur columns me. Agar kisi matrix A me m rows aur n columns hain, to uska order m × n (m by n) likha jaata hai — pehle rows, phir columns. Matrix ke elements ko aij se denote karte hain, jahan i = row number, j = column number.
| Symbol | Matlab |
|---|---|
| A = [aij]m×n | m rows, n columns wala matrix |
| a23 | 2nd row, 3rd column ka element |
| Total elements | m × n |
↔ Table ko side me swipe karein
2. Types of matrices
- Row matrix: sirf 1 row (order 1×n)
- Column matrix: sirf 1 column (order m×1)
- Square matrix: rows = columns (order m×m)
- Diagonal matrix: square matrix jisme saare non-diagonal elements zero hon
- Scalar matrix: diagonal matrix jisme saare diagonal elements EQUAL hon
- Identity matrix (I): scalar matrix jisme diagonal elements sab 1 hon
- Zero (null) matrix: saare elements zero
3. Matrix addition, subtraction, scalar multiplication
Do matrices tabhi add/subtract ho sakte hain jab dono ka order SAME ho — corresponding elements add/subtract hote hain. Scalar multiplication me har element ko ek number (scalar) se multiply karte hain.
4. Matrix multiplication
Matrix multiplication me AB possible hai sirf tabhi jab A ke columns ki sankhya B ke rows ke barabar ho — order match karna sabse pehli cheez check karni chahiye. Agar A order m×n hai aur B order n×p hai, to AB ka order hoga m×p. Element (AB)ij = A ki i-th row ka B ki j-th column ke saath dot product.
Important: generally AB ≠ BA (matrix multiplication commutative NAHI hai).
5. Transpose of a matrix
A ka transpose (AT ya A') A ke rows ko columns bana deta hai (aur vice versa). Agar A order m×n hai, to AT order n×m hoga.
6. Symmetric aur skew-symmetric matrices
- Symmetric: AT = A (yani aij = aji har i, j ke liye)
- Skew-symmetric: AT = −A (yani aij = −aji, aur diagonal elements hamesha zero hote hain)
Har square matrix A ko symmetric + skew-symmetric matrix ke sum me likha ja sakta hai: A = ½(A+AT) + ½(A−AT), jahan pehla part symmetric hai aur doosra skew-symmetric.
7. Elementary operations (brief)
Row/column par ye 3 operations allowed hain: (i) do rows/columns ko interchange karna, (ii) kisi row/column ko non-zero scalar se multiply karna, (iii) ek row/column me doosri row/column ka scalar multiple add karna. Inka use aage inverse nikalne me hota hai.
8. Invertible matrices (conceptual link)
Ek square matrix A "invertible" kehlata hai agar koi matrix B ho jispe AB = BA = I. Iska formal calculation (A-1 = adj(A)/|A|) next chapter Determinants me aayega — abhi bas concept samajhna hai ki har square matrix invertible nahi hota.
Poore Class 12 Maths ke handwritten colour notes
IITian & district toppers ke banaye short notes — revision-ready, diagram ke saath. Board se pehle poora syllabus 3 din me revise.
Exercise Questions — Solutions (Q1–Q18)
Q1. If a matrix has 12 elements, what are the possible orders it can have?
Order m×n aise ki m×n = 12. Possible orders: 1×12, 12×1, 2×6, 6×2, 3×4, 4×3.
Total elements = m × n = 12, so all factor pairs of 12 give valid orders.
Q2. Construct a 2×3 matrix A = [a_ij] whose elements are given by a_ij = (i + 2j).
A order 2×3 hoga.
a11=1+2=3, a12=1+4=5, a13=1+6=7
a21=2+2=4, a22=2+4=6, a23=2+6=8
A = [[3,5,7],[4,6,8]]
Q3. Find x and y if [[x+y, 2],[5, x-y]] = [[6, 2],[5, 2]].
Corresponding elements equal karo (same order matrices).
x+y = 6 ... (1)
x-y = 2 ... (2)
(1)+(2): 2x = 8 ⇒ x = 4
Put in (1): y = 6-4 = 2
Answer: x = 4, y = 2
Q4. If A = [[2,-1],[3,4]] and B = [[1,3],[0,2]], find A + B and A - B.
Same order (2×2), so element-wise add/subtract.
A+B = [[2+1, -1+3],[3+0, 4+2]] = [[3,2],[3,6]]
A-B = [[2-1, -1-3],[3-0, 4-2]] = [[1,-4],[3,2]]
Q5. If A = [[1,2],[3,4]], find 3A.
Scalar multiplication: har element ko 3 se multiply karo.
3A = [[3×1, 3×2],[3×3, 3×4]] = [[3,6],[9,12]]
Q6. Check whether AB is defined for A of order 3×2 and B of order 2×4. Find the order of AB if defined.
A columns (2) = B rows (2), so AB defined hai.
Order of AB = (rows of A) × (columns of B) = 3×4
Q7. If A of order 2×3 and B of order 2×3, is AB defined? What about A^T B?
A columns (3) ≠ B rows (2), so AB defined NAHI hai.
AT order = 3×2. AT columns (2) = B rows (2), so ATB defined hai, order 3×3.
Q8. If A = [[1,2],[3,4]] and B = [[2,0],[1,3]], find AB and BA. Show AB ≠ BA.
Dono 2×2 hain, so dono products defined hain.
AB = [[1×2+2×1, 1×0+2×3],[3×2+4×1, 3×0+4×3]] = [[4,6],[10,12]]
BA = [[2×1+0×3, 2×2+0×4],[1×1+3×3, 1×2+3×4]] = [[2,4],[10,14]]
Clearly AB = [[4,6],[10,12]] ≠ BA = [[2,4],[10,14]]
Q9. If A = [[1,2,3],[0,1,4]], find A^T. Also verify (A^T)^T = A.
A order 2×3, so AT order 3×2 (rows become columns).
AT = [[1,0],[2,1],[3,4]]
Taking transpose again: (AT)T = [[1,2,3],[0,1,4]] = A. Verified.
Q10. If A = [[2,-1],[3,4]] and B = [[0,1],[2,-3]], verify (A+B)^T = A^T + B^T.
Pehle A+B nikalo, phir transpose lo. Alag se AT + BT nikal ke compare karo.
A+B = [[2,0],[5,1]] ⇒ (A+B)T = [[2,5],[0,1]]
AT = [[2,3],[-1,4]], BT = [[0,2],[1,-3]]
AT+BT = [[2,5],[0,1]]
Both equal, so (A+B)T = AT+BT verified.
Q11. If A = [[1,2],[3,1]] and B = [[2,1],[0,1]], verify (AB)^T = B^T A^T.
Pehle AB nikalo, phir transpose. Alag se BTAT nikalo.
AB = [[1×2+2×0, 1×1+2×1],[3×2+1×0, 3×1+1×1]] = [[2,3],[6,4]]
(AB)T = [[2,6],[3,4]]
AT = [[1,3],[2,1]], BT = [[2,0],[1,1]]
BTAT = [[2×1+0×2, 2×3+0×1],[1×1+1×2, 1×3+1×1]] = [[2,6],[3,4]]
Both equal, so (AB)T = BTAT verified.
Q12. Show that A = [[3,5],[5,-2]] is a symmetric matrix.
Symmetric ke liye AT = A check karo.
AT = [[3,5],[5,-2]]
AT = A, so A symmetric hai.
Q13. Show that A = [[0,2,-3],[-2,0,4],[3,-4,0]] is a skew-symmetric matrix.
Skew-symmetric ke liye AT = -A check karo, aur diagonal zero honi chahiye.
Diagonal elements: 0, 0, 0 — sahi hai.
AT = [[0,-2,3],[2,0,-4],[-3,4,0]]
-A = [[0,-2,3],[2,0,-4],[-3,4,0]]
AT = -A, so A skew-symmetric hai.
Q14. Express A = [[3,5],[1,-1]] as the sum of a symmetric and a skew-symmetric matrix.
Formula: A = ½(A+AT) + ½(A-AT)
AT = [[3,1],[5,-1]]
A+AT = [[6,6],[6,-2]] ⇒ ½(A+AT) = [[3,3],[3,-1]] (symmetric part)
A-AT = [[0,4],[-4,0]] ⇒ ½(A-AT) = [[0,2],[-2,0]] (skew-symmetric part)
Check: [[3,3],[3,-1]] + [[0,2],[-2,0]] = [[3,5],[1,-1]] = A. Verified.
Q15. Find X and Y if X + Y = [[7,0],[2,5]] and X - Y = [[3,0],[0,3]].
Dono equations add aur subtract karke X, Y nikalo (jaise linear equations).
Adding: 2X = [[10,0],[2,8]] ⇒ X = [[5,0],[1,4]]
Subtracting: 2Y = [[4,0],[2,2]] ⇒ Y = [[2,0],[1,1]]
Q16. Find the value of x if 2[[1,3],[0,x]] + [[y,0],[1,2]] = [[5,6],[1,8]].
Pehle scalar multiply karo, phir add karo, phir corresponding elements barabar karo.
2[[1,3],[0,x]] = [[2,6],[0,2x]]
Adding: [[2+y,6],[1,2x+2]] = [[5,6],[1,8]]
2+y = 5 ⇒ y = 3
2x+2 = 8 ⇒ 2x = 6 ⇒ x = 3
Q17. If A = [[cos θ, sin θ],[-sin θ, cos θ]], verify that A^T A = I.
AT nikal ke A se multiply karo, aur cos²θ+sin²θ=1 identity use karo.
AT = [[cosθ, -sinθ],[sinθ, cosθ]]
ATA = [[cos²θ+sin²θ, -cosθsinθ+sinθcosθ],[sinθcosθ-cosθsinθ, sin²θ+cos²θ]]
= [[1,0],[0,1]] = I. Verified.
Q18. Find matrix X such that 2A + X = B, where A = [[1,2],[3,4]] and B = [[3,0],[1,2]].
X ko isolate karo: X = B - 2A
2A = [[2,4],[6,8]]
X = [[3,0],[1,2]] - [[2,4],[6,8]] = [[1,-4],[-5,-6]]
Important Equations — Ek Nazar Me
| Concept | Definition / Formula |
|---|---|
| Order of matrix | m rows × n columns = m × n |
| Diagonal matrix | square matrix, non-diagonal elements = 0 |
| Scalar matrix | diagonal matrix with all diagonal elements equal |
| Identity matrix I | scalar matrix with diagonal elements = 1 |
| Addition/Subtraction | only defined when both matrices have same order |
| Multiplication AB | defined only when columns of A = rows of B; order of AB = (rows of A) × (columns of B) |
| Commutativity | AB ≠ BA in general |
| Transpose sum rule | (A+B)T = AT + BT |
| Transpose product rule | (AB)T = BTAT (order reverses) |
| Transpose of transpose | (AT)T = A |
| Symmetric matrix | AT = A |
| Skew-symmetric matrix | AT = -A (diagonal elements always 0) |
| Sum decomposition | Any square matrix A = ½(A+AT) + ½(A-AT), symmetric + skew-symmetric parts |
| Invertible matrix | A is invertible if ∃ B such that AB = BA = I |
↔ Table ko side me swipe karein
Common Mistakes — Yahan Marks Kat te Hain
- AB ko BA ke barabar maan lena. Matrix multiplication commutative nahi hai — AB aur BA generally alag hote hain, hamesha alag alag calculate karo.
- Order match kiye bina multiplication try karna. AB tabhi defined hai jab A ke columns = B ke rows. Pehle order likh ke check karo, warna galat answer aayega.
- Addition/subtraction me bhi order check na karna. Sirf same order ke matrices hi add/subtract ho sakte hain — different order matrices ko add karna hi galat hai.
- Symmetric-skew-symmetric decomposition me sign ki galti. Symmetric part = ½(A+AT) aur skew-symmetric part = ½(A-AT) — in dono ko mix mat karo, sign carefully lagao.
- (AB)^T ko A^T B^T likh dena. Sahi rule hai (AB)T = BTAT — order REVERSE hota hai, students yahi bhool jaate hain.
- Skew-symmetric matrix ke diagonal elements ko check na karna. Skew-symmetric matrix me diagonal elements hamesha zero hone chahiye (kyunki aii = -aii tabhi possible hai jab aii=0) — proof me ye step bhool jaana common mistake hai.
Board-Style Important Questions
- 1 mark: If a matrix has 8 elements, write all possible orders it can have.
- 1 mark: If A is a matrix of order 3×2, write the order of A^T.
- 2 marks: Construct a 2×2 matrix A = [a_ij] whose elements are given by a_ij = (i-j)^2.
- 3 marks: If A = [[2,3],[1,4]] and B = [[1,0],[2,-1]], find AB and BA. Verify AB ≠ BA.
- 3 marks: Express the matrix [[2,-2],[-1,3]] as the sum of a symmetric and a skew-symmetric matrix.
- 5 marks: If A = [[cos θ, sin θ],[-sin θ, cos θ]], verify that A^T A = I, where I is the identity matrix of order 2.
Aksar Poochhe Jaane Wale Sawaal
Matrix aur determinant me kya farak hai?
Matrix ek rectangular array hai numbers ka jispe apni algebra hoti hai (addition, multiplication). Determinant sirf square matrix se associated ek single NUMBER hota hai, jo agle chapter me padhoge.
AB defined hai to kya BA bhi hamesha defined hoga?
Nahi zaroori nahi. AB defined hone ke liye A ke columns = B ke rows chahiye. BA defined hone ke liye B ke columns = A ke rows chahiye — dono conditions alag hain, dono ek saath sach nahi hoti hamesha.
Kya har matrix ka transpose exist karta hai?
Haan, har matrix ka transpose exist karta hai, chahe woh square ho ya nahi. Bas order swap ho jaata hai — m×n matrix ka transpose n×m ho jaata hai.
Skew-symmetric matrix ke diagonal elements hamesha zero kyun hote hain?
Kyunki skew-symmetric matrix me a_ij = -a_ji hota hai. Diagonal par i=j hota hai, to a_ii = -a_ii, jo tabhi possible hai jab a_ii = 0.
Kya zero matrix symmetric aur skew-symmetric dono ho sakta hai?
Haan, zero matrix (jisme saare elements zero hon) dono conditions satisfy karta hai — A^T = A bhi true hai aur A^T = -A bhi true hai, isliye ye ek exception case hai.
Matrix ka inverse har matrix ka possible hota hai kya?
Nahi, sirf square matrices ka inverse possible ho sakta hai, aur woh bhi tabhi jab matrix invertible ho (non-singular). Iski detailed condition next chapter Determinants me padhoge.
Class 12 Maths — Saare Chapters
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