Matrices — review, operations, types: Worked examples

4.5.6 · D3 · Maths › Linear Algebra (Full) › Matrices — review, operations, types

Scenario matrix

Har matrix exercise inme se kisi ek box mein aati hai. Agar tum har box se ek example kar sako, toh exam mein kuch bhi naya nahi hoga.

Cell	Scenario	Kya mushkil hai
A	Compatible product $A B$ , dono nonzero	row·column pairing sahi karna
B	Non-commutativity: $A B$ aur $B A$ compute karo	ye dikhana ki dono alag hain
C	Degenerate: zero matrix / identity se multiply karna	"kuch nahi karta" aur "collapse" wale cases
D	Zero divisors: $A B = 0$ with $A \neq = 0, B \neq = 0$	woh surprising box jo number-intuition tod deta hai
E	Transpose reversal $(A B)^{⊤} = B^{⊤} A^{⊤}$	order flip ho jaata hai
F	Symmetric + skew split	$\frac{1}{2}$ trick, zero diagonal
G	Shape mismatch: jab product exist hi nahi karta	dimensions padhna
H	Limiting / repeated: powers $A^{2}, A^{3}$ of a matrix	khud ke saath composition
I	Word problem: matrices se kuch real cheez model karna	words ko grid mein translate karna
J	Exam twist: orthogonal matrix, verify $A^{⊤} A = I$	length-preserving check

Neeche har symbol scratch se banaya gaya hai. Ek matrix bas ek rectangular grid of numbers hai; ek entry $a_{ij}$ row $i$ , column $j$ mein hoti hai (rows upar se neeche, columns left se right). Ek row aur ek column ka dot product matlab: matching numbers ko multiply karo aur add karo — yahi ek idea is page ke har product ko power karta hai. Ise ek horizontal strip ko ek vertical strip pe slide karte hue socho:

Figure — Matrices — review, operations, types

Cell A — ek compatible product

Worked example

$A B$ compute karo $A = [20 - 1 3], B = [12 4 - 2]$

Forecast: aage padhne se pehle, top-left entry guess karo. Ye $A$ ki row 1 = $(2, - 1)$ aur $B$ ki column 1 = $(1, 2)$ ka dot product hai. Apna guess yaad rakho.

Step 1 — shapes check karo. Ye step kyun? Product tab hi exist karta hai jab inner dimensions match karein: $A$ is $2 \times 2$ , $B$ is $2 \times 2$ , inner numbers dono $2$ hain. ✓ Result $2 \times 2$ hoga.

Step 2 — entry $(1, 1)$ : row 1 · col 1. $2 \cdot 1 + (- 1) \cdot 2 = 2 - 2 = 0$ Ye step kyun? Har output entry mein ek row aur ek column milte hain; kuch aur combine nahi hota.

Step 3 — baaki entries. $(1, 2) : 2 \cdot 4 + (- 1) (- 2) = 8 + 2 = 10$ $(2, 1) : 0 \cdot 1 + 3 \cdot 2 = 6$ $(2, 2) : 0 \cdot 4 + 3 \cdot (- 2) = - 6$ $A B = [06 10 - 6]$

Verify: $(1, 1)$ entry $0$ aayi — yahan ek coincidence hai, koi rule nahi. Sanity check: $A$ ki row 2 mein $0$ se shuru hota hai, isliye $A B$ ki row 2 ki dono entries sirf $3$ se aati hain: $3 \cdot (2, - 2) = (6, - 6)$ . ✓ Match karta hai.

Cell B — order matters ( $A B \neq = B A$ )

Cell C — degenerate multipliers: $I$ aur $0$

Cell D — zero divisors (surprising box)

Cell E — transpose reversal $(A B)^{⊤} = B^{⊤} A^{⊤}$

Cell F — symmetric + skew split

Cell G — mismatch (product exist nahi karta)

Cell H — powers (khud ke saath composition)

Cell I — ek real-world word problem

Cell J — exam twist: orthogonality verify karo

Worked example Kya ye rotation matrix orthogonal hai?

$R = [cos θ sin θ - sin θ cos θ]$ Ek matrix orthogonal hoti hai jab $R^{⊤} R = I$ ; aisi maps lengths preserve karti hain — pure rotations/reflections (Eigenvalues and Eigenvectors mein central hain).

Forecast: expand karne se pehle $R^{⊤} R$ guess karo — kya wo $I$ hona chahiye?

Step 1 — $R^{⊤} R$ banao. $R^{⊤} = [cos θ - sin θ sin θ cos θ]$ . $(1, 1) : cos^{2} θ + sin^{2} θ, (1, 2) : - cos θ sin θ + sin θ cos θ, \dots$ Ye step kyun? Har entry $R$ ke do columns ka dot product hai; orthogonality ka matlab hai columns unit-length hain aur perpendicular hain.

Step 2 — $cos^{2} θ + sin^{2} θ = 1$ use karke simplify karo. $R^{⊤} R = [1001] = I$

Verify (numeric, $θ = 3 0^{\circ}$ ): $cos 3 0^{\circ} = \frac{3}{2}, sin 3 0^{\circ} = \frac{1}{2}$ . Toh $(1, 1) = \frac{3}{4} + \frac{1}{4} = 1$ ✓ aur off-diagonal $- \frac{3}{2} \cdot \frac{1}{2} + \frac{1}{2} \cdot \frac{3}{2} = 0$ ✓. Rotation length preserve karta hai jaise expected tha.

Recall Kaun sa cell kaun sa tha?

Har product tab exist karta hai jab ::: inner dimensions match karein (pehle matrix ke columns = doosre matrix ke rows). Woh box jo number-intuition tod deta hai ::: zero divisors, $P Q = 0$ with $P, Q \neq = 0$ (Cell D). Ek product ka transpose ::: order reverse karta hai: $(A B)^{⊤} = B^{⊤} A^{⊤}$ (Cell E). Skew part mein hamesha hoti hai ::: zero diagonal (Cell F). Shear $[1011]$ ke liye $A^{n}$ ::: $[10 n 1]$ (Cell H). Orthogonal test ::: $R^{⊤} R = I$ (Cell J).

Connections

Matrices — review, operations, types — woh parent note jise ye examples deepen karte hain.
Dot Product and Vectors — har product entry inme se ek hai (Cells A, I).
Linear Transformations — repeated maps ke roop mein powers aur shears (Cell H).
Determinant and Inverse of a Matrix — zero divisors cancellation kyun block karte hain (Cell D).
Eigenvalues and Eigenvectors — orthogonal matrices dobara aati hain (Cell J).
Systems of Linear Equations — woh compression jo ye sab motivate karta hai.