2.6.1 · D5 · HinglishMatrices & Determinants — Introduction
Question bank — Matrix definition — rows, columns, order, elements
2.6.1 · D5· Maths › Matrices & Determinants — Introduction › Matrix definition — rows, columns, order, elements
Shuru karne se pehle, ek do-line ka reminder un vocabulary words ka jo har question mein use hoti hain, taaki koi symbol unfamiliar na lage:
Sach ya jhooth — justify karo
Ek matrix aur ek matrix kabhi bhi equal nahi ho sakti, chahe unme koi bhi numbers hon
Sach — equality pehle identical order maangti hai, aur alag shapes hain, toh comparison numbers dekhne se pehle hi fail ho jaata hai.
Order mein ordinary multiplication jaisa hi rule hota hai, toh aur ek hi shape describe karte hain
Jhooth — "order" mein ek label hai, multiplication nahi; yaani 4 rows ki height, yaani 5 rows ki height, toh ye genuinely alag grids hain.
Har square matrix ek rectangular arrangement of numbers bhi hoti hai
Sach — square sirf woh special rectangle hai jisme width aur height barabar ho; "rectangular array" woh umbrella hai jiske andar square matrices aati hain.
12 elements wali matrix ka order zaroor hoga
Jhooth — total elements ke kai factorisations hain (), toh sirf count se shape fix nahi hoti.
element hamesha ek matrix mein exist karta hai
Jhooth — row index 3 available 2 rows se zyada hai, toh kisi cheez ki taraf point nahi karta; ek index tabhi valid hai jab aur .
Ek matrix brackets mein likha hua ek plain number hi hai
Ek object ke roop mein sach hai, lekin ise order ki matrix ki tarah treat kiya jaata hai; brackets matter karte hain kyunki ab ye matrix operations mein part le sakti hai, ek loose scalar ki tarah nahi.
Element mein, par swap karna generally ek alag entry par point karta hai
Sach — jab tak na ho (ek diagonal element), row--column- padhna versus row--column- padhna do alag grid cells par land karta hai; yahi swap transpose poori matrix ke saath karta hai.
Har row matrix ek column matrix bhi hoti hai
Jhooth — row matrix hoti hai (ek horizontal strip) aur column matrix hoti hai (ek vertical strip); ye sirf ke single case mein ek saath milti hain.
Error pakdo
" ka matlab hai column 2, row 3 mein entry." Galti dhundho
Do errors hain: convention row pehle, column baad mein hai, toh row 2, column 3 hai — reader ne dono indices ki roles ulti kar di.
Ek student ka order likhta hai. Kya galat hua?
Unhone teen numbers ko "rows" ke roop mein count kiya. Ye teen columns wali ek horizontal row hai, toh order hai; entries count karna rows count karna nahi hai.
"Kyunki indices zero se start hoti hain, top-left entry hai." Ise sahi karo
Standard mathematical notation 1-based hai, toh top-left entry hai; zero-start ki aadat programming arrays se aati hai, matrix algebra se nahi.
Koi claim karta hai ki square hai kyunki "iske 6 elements hain jaise ek ke." Flaw kahan hai?
"Square" shape ke baare mein hai (), element count ke baare mein nahi; yeh matrix hai jisme , toh ye rectangular hai chahe isme kitne bhi numbers hon.
banane ke liye jisme ho, ek student compute karta hai. Kya value sahi hai, aur kya reasoning safe hai?
Value yahan sahi hai, lekin sirf isliye kyunki symmetric hai; reasoning unsafe hai — unhone use kiya ki jagah, jo kisi bhi non-symmetric formula jaise ke liye galat answer deta.
"Ek matrix aur ek matrix automatically equal hain kyunki unka order same hai." Ise theek karo
Equal order necessary but not sufficient hai; matrices ko element-by-element bhi match karna chahiye, har ke liye, tab hi hum unhe equal keh sakte hain.
Why questions
Hum par ek running number ki jagah do subscripts kyun use karte hain?
Ek grid two-dimensional hoti hai, toh ek coordinate ek cell ko pin nahi kar sakta; kisi bhi entry ko unambiguously locate karne ke liye hume (which row, which column) — do numbers — chahiye.
Convention "row pehle, column baad mein" kyun hai, ulte kyun nahi?
Ye is baat se mirror karta hai ki hum order (rows before columns) kaise likhte hain aur general form kaise laid out hoti hai, har jagah ek consistent left-to-right, top-to-bottom reading rule rakhta hai.
Matrix ka order add ya multiply karne se pehle kyun matter karta hai?
Order ek "shape passport" hai: addition ke liye identical orders chahiye aur multiplication ke liye inner dimensions match karni chahiye, toh shape pehle se decide karti hai ki kaun se operations legal bhi hain.
Linear algebra mein ek point likhne ka conventional tarika column vector kyun hai, row vector kyun nahi?
Kyunki transformations matrix-times-column, , ke roop mein apply hoti hain; point ko column ke roop mein likhna multiplication ko right side par dimensionally line up karta hai.
Ek matrix expressions (jaise ya ) kaise rakh sakti hai aur phir bhi matrix count hoti hai?
Definition sirf numbers ya expressions ki rectangular arrangement maangti hai; grid structure hi ise matrix banati hai, ye nahi ki har cell pehle se bare number tak simplified ho.
Hum ek system of equations ko ek matrix mein pack karne ki taklif kyun karte hain?
Ek akeli matrix hume ek set of systematic rules (determinant, row operations, solving systems) sabhi equations par ek saath apply karne deti hai, har equation ko alag-alag juggle karne ki jagah.
Edge cases
Kya ek matrix mein zero rows ya zero columns ho sakte hain?
Standard school-level treatment mein, nahi — ek matrix mein kam se kam ek row aur ek column honi chahiye (); ek "empty" grid mein koi elements nahi hote aur yahan use nahi hoti.
Sabse chhoti possible matrix kya hai, aur uska order kya hai?
Brackets mein ek akela number, order ; ye ek saath sabse chhoti square, row, aur column matrix hai.
Agar har aur ke liye ho, toh kya ye phir bhi valid matrix hai?
Haan — zero matrix bilkul valid hai; "matrix" arrangement describe karta hai, aur koi rule nahi hai jo entries ko sab zero hone se roke (dekho types of matrices).
Ek square matrix mein kitne elements satisfy karte hain?
Exactly — ye diagonal elements hain jo top-left se bottom-right tak jaate hain.
Ek row matrix mein, pehla index kab bhi kitna hoga?
Hamesha , kyunki sirf ek hi row hai; sirf column index hi se tak vary karta hai, isliye practice mein row vectors ko aksar ek single subscript ke saath likha jaata hai.
Kab aur guaranteed same cell hoge, matrix ki values se independent?
Sirf jab ho (ek diagonal position); diagonal se bahar ye alag cells hain jinki values same ho bhi sakti hain ya nahi bhi.
Kisi row mein zyada elements add karne se matrix ka order badal jaata hai?
Haan — ek row ko wide karna column count badhata hai, toh ek matrix ban jaati hai; tum element counts nahi badal sakte order badlaaye bina.