Types of matrices — row, column, square, diagonal, identity, zero, symmetric, skew-symmetric: Exercises

2.6.2 · D4 · Maths › Matrices & Determinants — Introduction › Types of matrices — row, column, square, diagonal, identity,

Yeh page ek self-test ladder hai. Har problem ka fully worked solution ek collapsible callout mein chhupa hua hai — problem padho, paper par try karo, phir reveal karo. Jo bhi symbols hum use karte hain ( $A^{T}$ , "diagonal", "order", $diag$ , $\dots$ ) — yeh sab parent note Types of Matrices mein define hain; agar doubt ho toh wahan wapas jao. Hum Matrix Transpose aur Matrix Operations par bhi rely karte hain.

Dono symmetry words ka ek quick reminder, kyunki aadhe exercises unhi par hinge karte hain:

Neeche wali picture "diagonal ke across mirror" ka woh mental model hai jise hum baar baar use karenge.

Figure — Types of matrices — row, column, square, diagonal, identity, zero, symmetric, skew-symmetric

Level 1 — Recognition

Problem 1.1

Har matrix ka type name karo (sabse specific correct naam do): $P = [401], Q = [0000], R = [50 0 - 2], S = [1001]$

Recall Solution 1.1

KYA check karte hain: pehle shape, phir zero-pattern.

$P$ mein ek row hai, teen columns hain (order $1 \times 3$ ) → row matrix.
$Q$ square hai aur har entry $0$ hai → zero matrix (technically diagonal bhi hai, lekin "zero" zyada specific hai).
$R$ square hai, saare off-diagonal entries $0$ hain, aur diagonal entries $5, - 2$ sab $1$ nahin hain → diagonal matrix (identity nahin).
$S$ diagonal hai aur saare diagonal entries $= 1$ hain → identity matrix $I_{2}$ . "Most specific" kyun: identity ek special diagonal hai, aur zero ek special sab-kuch hai — hamesha sabse tight pattern naam karo.

Problem 1.2

Inme se kaun square hain? Har ek ka order batao. $T = [29], U = [1331], V = [142536]$

Recall Solution 1.2

Ek matrix square tab hota hai jab rows $=$ columns.

$T$ : order $2 \times 1$ (2 rows, 1 column) → column matrix, square nahin.
$U$ : order $2 \times 2$ → square (aur symmetric bhi, kyunki $a_{12} = 3 = a_{21}$ ).
$V$ : order $2 \times 3$ → square nahin. Sirf $U$ square hai.

Level 2 — Application

Problem 2.1

$A^{T}$ compute karo aur batao ki $A$ symmetric hai, skew-symmetric hai, ya na kuch bhi. $A = [0 - 7 70]$

Recall Solution 2.1

KYA: rows↔columns flip karo. $A$ ka Row $1$ hai $(0, 7)$ ; yeh $A^{T}$ ka column $1$ ban jaata hai. $A^{T} = [07 - 7 0]$ $- A$ se compare kyun karte hain: $- A = [07 - 7 0]$ . Yeh exactly $A^{T}$ ke barabar hai, toh $A^{T} = - A$ . Diagonal bhi saari zeros hai — skew rule ke saath consistent hai. $A$ skew-symmetric hai.

Problem 2.2

$x, y, z$ fill karo taaki matrix symmetric ho: $M = 25 z x y 4 - 1 43$

Recall Solution 2.2

Rule used: symmetric ka matlab $a_{ij} = a_{j i}$ (diagonal ke across mirror — figure dekho).

$a_{21} = 5$ ka mirror $a_{12} = x$ hai, toh $x = 5$ .
$a_{31} = z$ ka mirror $a_{13} = - 1$ hai, toh $z = - 1$ .
$a_{23} = 4$ aur $a_{32} = 4$ already match karte hain ✓.
$y$ diagonal par hai, toh yeh apne aap se mirror hota hai; symmetry iske liye koi constraint nahin lagaati — koi bhi value chalega. $y$ free lo (jaise $y = 0$ ). Answer: $x = 5, z = - 1, y$ arbitrary.

Problem 2.3

$D = diag (3, - 1, 2)$ diya gaya hai, $D^{2}$ aur $D I_{3}$ compute karo.

Recall Solution 2.3

Diagonal powers easy kyun hote hain: diagonal matrices ko multiply karna sirf matching diagonal entries ko multiply karna hai (off-diagonals $0$ rehte hain). Toh $D^{k} = diag (d_{1}^{k}, d_{2}^{k}, d_{3}^{k})$ . $D^{2} = diag (3^{2}, (- 1)^{2}, 2^{2}) = diag (9, 1, 4) = 900010004$ Aur $D I_{3} = D$ kyunki identity multiplicative "1" hai: $D I_{3} = 300 0 - 1 0 002$ .

Level 3 — Analysis

Problem 3.1

Kya koi matrix dono symmetric aur skew-symmetric ho sakta hai? Agar haan, toh aisi har matrix describe karo.

Recall Solution 3.1

Dono conditions ek saath set karo. Symmetric deta hai $a_{ij} = a_{j i}$ ; skew deta hai $a_{ij} = - a_{j i}$ . Combine karo: $a_{j i} = - a_{j i}$ , toh $2 a_{j i} = 0$ , yani saare $i, j$ ke liye $a_{ij} = 0$ . Conclusion: woh ek hi matrix jo dono hai woh zero matrix $O$ hai. (Isliye zero matrix quietly almost har symmetry definition satisfy karta hai.)

Problem 3.2

Ek matrix $B$ skew-symmetric hai aur $3 \times 3$ hai. Diagonal ke upar ke entries hain $b_{12} = a, b_{13} = b, b_{23} = c$ . Saare $9$ entries likho, phir count karo ki ek $3 \times 3$ skew-symmetric matrix mein kitne independent numbers hote hain.

Recall Solution 3.2

Diagonal entries $0$ par force ho jaate hain. Diagonal ke neeche ke entries upar ke mirror ke negatives hain: $B = 0 - a - b a 0 - c b c 0$ Freedom count karna: sirf teen above-diagonal entries $a, b, c$ free hain; baaki sab unse determine ho jaate hain. Toh ek $3 \times 3$ skew-symmetric matrix mein $3$ independent entries hote hain. (General rule: $\frac{n ( n - 1 )}{2}$ ; $n = 3$ ke liye yeh $\frac{3 \cdot 2}{2} = 3$ hai ✓.)

Problem 3.3

Kya $[1223]$ diagonal hai? Kya yeh symmetric hai? Apne answer se jo relationship reveal hoti hai woh explain karo.

Recall Solution 3.3

Diagonal? Nahin — off-diagonal entry $2 \neq = 0$ hai.
Symmetric? Haan — $a_{12} = 2 = a_{21}$ . Revealed relationship: har diagonal matrix symmetric hota hai, lekin har symmetric matrix diagonal nahin hota. Diagonal, symmetric ka ek strict subset hai (dekho the parent's mistake callout).

Level 4 — Synthesis

Problem 4.1

$A = [4268]$ ko symmetric part $S = \frac{1}{2} (A + A^{T})$ aur skew part $K = \frac{1}{2} (A - A^{T})$ mein split karo. Verify karo ki $S + K = A$ .

Recall Solution 4.1

Yeh kyun kaam karta hai: $\frac{1}{2} (A + A^{T})$ hamesha symmetric hota hai (ek matrix ko uske transpose ke saath add karna use mirror karta hai), aur $\frac{1}{2} (A - A^{T})$ hamesha skew hota hai (subtract karne par diagonal cancel ho jaata hai). Unka sum telescope karke wapas $A$ de deta hai. Pehle $A^{T} = [4628]$ . $S = \frac{1}{2} [88816] = [4448], K = \frac{1}{2} [0 - 4 40] = [0 - 2 20]$ Check: $S + K = [4268] = A$ ✓. Note karo $S$ symmetric hai, $K$ skew hai (zero diagonal).

Problem 4.2

Maano $A$ koi bhi square matrix hai. Prove karo ki $A + A^{T}$ symmetric hai.

Recall Solution 4.1... (4.2)

Tool: do transpose rules — $(X + Y)^{T} = X^{T} + Y^{T}$ aur $(X^{T})^{T} = X$ . Maano $B = A + A^{T}$ . Toh $B^{T} = (A + A^{T})^{T} = A^{T} + (A^{T})^{T} = A^{T} + A = A + A^{T} = B .$ Kyunki $B^{T} = B$ , matrix $B$ symmetric hai. $■$

Level 5 — Mastery

Problem 5.1

Prove karo ki har square matrix $A$ ko uniquely symmetric $+$ skew-symmetric ke roop mein likha ja sakta hai.

Recall Solution 5.1

Existence. Set karo $S = \frac{1}{2} (A + A^{T})$ aur $K = \frac{1}{2} (A - A^{T})$ . Problem 4.2 ke argument se, $S^{T} = S$ (symmetric) aur $K^{T} = \frac{1}{2} (A^{T} - A) = - K$ (skew). Aur $S + K = A$ . Toh ek decomposition exist karta hai. Uniqueness. Maano $A = S^{'} + K^{'}$ jahan $S^{'}$ symmetric hai, $K^{'}$ skew hai. Transpose karo: $A^{T} = S^{' T} + K^{' T} = S^{'} - K^{'}$ . Dono equations add aur subtract karo: $A + A^{T} = 2 S^{'} \Rightarrow S^{'} = \frac{1}{2} (A + A^{T}) = S, A - A^{T} = 2 K^{'} \Rightarrow K^{'} = \frac{1}{2} (A - A^{T}) = K .$ Toh pieces force ho jaate hain — decomposition unique hai. $■$

Problem 5.2

Prove karo ki kisi bhi $n \times n$ skew-symmetric matrix $B$ ke liye jahan $n$ odd ho, trace (diagonal entries ka sum) $0$ hota hai, aur batao ki yeh immediately kyun obvious hai.

Recall Solution 5.2

Immediate reason: skew-symmetry har $i$ ke liye $b_{ii} = 0$ force karta hai ( $b_{ii} = - b_{ii}$ se). Trace hai $b_{11} + b_{22} + \dots + b_{nn} = 0 + 0 + \dots + 0 = 0$ . Yeh sabhi $n$ ke liye hold karta hai, odd ya even — "odd" hint ek distractor hai. Kisi bhi skew-symmetric matrix ka trace $0$ hota hai.

Problem 5.3

Maano $D = diag (2, 3, 5)$ . Easy-power rule use karke $D^{4}$ ka diagonal compute karo.

Recall Solution 5.3

$D^{4} = diag (2^{4}, 3^{4}, 5^{4}) = diag (16, 81, 625)$ . Isliye Diagonalization jaise methods itne valued hain: jaise hi koi matrix diagonal form mein aa jaaye, powers (aur isliye eigenvalue-based analysis) ek-line computations ban jaati hain.

Recall Self-test summary

Har level ka ek-line takeaway ::: L1 sabse tight pattern naam karo; L2 transpose = $(A^{T})_{ij} = A_{j i}$ ; L3 skew diagonals $0$ hote hain; L4 har part ko aadha karo; L5 har square matrix = unique symmetric + skew. Woh matrix jo dono symmetric aur skew-symmetric hai ::: zero matrix $O$ . $3 \times 3$ skew matrix mein independent entries ::: $3$ (yani $\frac{n ( n - 1 )}{2}$ ).