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Made with Xodo PDF Reader and Editor Made with Xodo PDF Reader and Editor Topics To Be Covered Definition and Order of a Matrix Algebra of Matrices and Trace of a Matrix Symmetric and Skew Symmetric Matrix 4& talel Adjoint and Inverse of A Matrix O N E S H OT Physics Wallah By- Tarun Khandelwal sir ( K. Sir) B-Tech IIT Delhi, (AIR Matrix Method for Solving Linear Equations ”* Characteristic Equation and Cayley Hamilton Theorem * * Made with Xodo PDF Reader and Editor Made with Xodo PDF Reader and Editor Matrix Definition ORDER ( Je are Rows no-ef columS A matrix is Arrangement of Some numbers/Elements in a Rectangular array. Ud Unlike a Determinant, A matrix has no value. ed A= Jc d| isa Matrix of Orderx2) A~-f@ b ¢ A e f of | ’y — C ( (x4) Az=Jd f| is a Matrix of Order__( 2x3) g h L = &) Jobvevy aA=|i 2 3 HW is a Matrix of Order 2%“ > 45 Z2 5 1 —___ a G6 0) % & (Qx3) SQUARE Matai Goren) Made with Xodo PDF Reader and Editor wv” a ee. Se Ag Aioganol ef ordy Ag medix Ex st ne Papanepmaprwmmerwanrrarararacananancney 1. 2. QA\o Ko daagorol Made with Xodo PDF Reader and Editor QUESTIONS i Write all possible order if given matrix has: On (i) (ii) 21 elements (aixl) Ov (1x21) 03 4x3) w (3x7) | 36 elements @éx\), (\x36) i'bx2) Qx18) ' (3x12) ev (12%3) (4X3) (9x4) (6x 6) oro —\ Ka —— @ G Arar - an} Made with Xodo PDF Reader and Editor Matrix Definition- Summary A matrix is arrangement of Some numbers / Elements in a Rectangular array. denotes its column number. A matrix has no value, unlike a Determinant. Determinant value of only a Square Matrix Exists. Sa ee ——” a fm = n, then matrix is called a square matrix. Determinant of a square matrix A is denoted by |A\. Made with Xodo PDF Reader and Editor Vp N @ QUESTION Number of different matrices of order 3 x 3 can be formed with help of four 0's and five 2's. ae Any element of a matrix A is of the form a; where “i’ denotes row number and “j fa matrix has m rows and n columns, then its order is (m x n). —~ USING ol of Hum 0, 0,0,0, 2,2,2,2,2 oo) oO oO 4 6x3) aQ0|”" 0 COO Crs CxSXS) @ @ Made with Xodo PDF Reader and Editor @ QUESTION [Ans. (A) > False Made with Xodo PDF Reader and Editor = I Y Types of Matrices Check following statements as True or False: (B) — False (C) > True] (7) Upper Triangular Matrix: A Matrix in which all the elements below leading (A) If |A] = 0 => Ais null matrix (A=0). ¢ F) diagonal are necessarily zero. (B) If‘A’ is Null matrix (A= 0) > |A]=0. ( F) i DAT eee S (C)| If ‘A’ is Square Null matrix > |A| = 0. Gy ) (Ss| 2 & |A|=o but A Is nok ntl (8) Lower Triangular Matrix: A Matrix in which all the elements above leading A= 2, ‘ diagonal are necessarily zero. SK2 Made with Xodo PDF Reader and Editor |} 0 ° Types of Matrices & Fs >) (9) Diagonal Matrix: A matrix in which all the Non diagonal Elements are necessarily zero. < (Ff) (C) Every Null Matrix is a Triangular Matrix. C F) ——_ D) Ina Horizontal Matrix Number of Rows is = 1. (F) ————__ F) Determinant value of Every Matrix Exists. ( F) Nee eee Lo” €-% (-— S £ LEN H) Every Diagonal Matrix is a Triangular Matrix. (T) Made with Xodo PDF Reader and Editor oo? BRAIN TEASER Hea A is a square matrix of order n. | @ (S29 (A) In an upper Triangular Matrix, all the elements Ebovd diagonal are Necessarily (B) Ina Triangular matrix, diagonal elements are necessarily non-zero. C & ) So 9 Oo 90°90 E) Ina Horizontal Matrix Number of Rows is more than the number of Columns. (fF) G) A Scalar matrix is both upper Triangular as well as lower Triangular Matrix. (T) @ NOTE THAT . (Minimum numberof zeros in a diagonal matrix of order n is ———_— We NAKA ee | e Minimum number of zero in a triangular matrix of order n is (tn) (Joxto) Uppr ® Ron motry 2 lo—lo — a” Algebra of Matrices p = minimum number of zeroes if A is a triangular matrix Equality of Matrices ! = maximum number of distinct entries if A is a triangular matrix Addition/ subtraction of matrices If! + 5 = p + 2m, find the order of the matrix. — Yat (in =e 2 gp NS pi Multiplication with a scalar Multiplication of 2 matrices Y Made with Xodo PDF Reader and Editor @ TN —W SSS mea min Kitna Zanror ( Gs) loo-lo _ Made with Xodo PDF Reader and Editor @ clivisten 1S pot df Made with Xodo PDF Reader and Editor Made with Xodo PDF Reader and Editor Multiplication of a Matrix with a Scalar If a matrix A is multiplied by a non-zero Scalar “k”, then each element of the matrix gets multiplied with “k’. : b\)_ (2a 2b Cc djJ~[ 2c 2d ka Kb Ex) Ke KF a b c ad e f ¢,. (ax (249 ax2 E Az 2x2 aolb ie OPtbry AAbs Bo QW2 A CH) L* a cq+d5 BA AZAD ES & (°) _ | Parre Pb+%4 but QB + BA). ee _1Am+sa Ab+sd Made with Xodo PDF Reader and Editor Matrix Multiplication e Multiplication AB exists only when number of columns in A = number of Rows in B j e Iforder of Ais (m x n) and order of B is (n x p), then order of AB is (m x p) e IfAandB are two matrices such that: (i) | AB = BA then A and B are said to commute. | (ii) AB = —BA then A and B are said to anti-commute. Made with Xodo PDF Reader and Editor | QUESTION [Fane ‘ S|. then ao 31 is equal to: _ -4 oe esis @ @ ji 0) (1 2026: 4) 2025 F QUESTION ae ifA=@ 2 3)andB = c f (“2 = ©00 © (2026 0] 12026 Made with Xodo PDF Reader and Editor @ Z : , then AB = — (1x2) (3x3) = (x2, a 0 4 -7 Leh lee) (e Pee 4+ 2 24-3x-3 att 4] Ab= Made with Xodo PDF Reader and Editor QUESTION lffA= [| " , then A2°° js equal to: @ aRtalanap le 2 See (Cdk 1 vn \ AY. |) 1 = il 0] 1 Properties of Matrix Multiplication Made with Xodo PDF Reader and Editor Made with Xodo PDF Reader and Editor (a) Matrix multiplication is not Commutative i.e., (b) (c) (d) AB = 0 SEitherA = OorB=0 (in general) Matrix Multiplication is Associative: If A,B and C are conformable for the product AB and BC, then AB # BA Matrix Multiplication is Distributive over Addition A(B + C) = AB+ AC (A+B)C = AC+BC rovided A, B and C are conformable (AB)C = A(BC) e) If A & B are two square matrices then (A + B)? # (A2+ B2+ 2AB) | ® = If Ais a square matrix & | is Identitiy then (I + A)? = (I + A? + 2A) f) q ¥ ~ Made with Xodo PDF Reader and Editor @ @ QUESTION & [Ans. A] Let A = [ais],.9° where aj; # 0 for all i,j and At =], Let a be the sum of all diagonal elements of A and b = |A| Then 3a? + 4b? is equal to: me (LS Made with Xodo PDF Reader and Editor ) @ 2 areas 2 7 = A+8) = B > (A+B): (H+8)(A+8) = Ar AB+ BA ~ Lh RKB Commute (46-64) Tan Greys AX S420B . (+A) = D+ A 2rA = Ith 4 8 m SS (Itay et ve a Made with Xodo PDF Reader and Editor Made with Xodo PDF Reader and Editor More about Matrix Multiplication Tf RUB are Sonor QUESTION ep uf) ictal If A, B and C are matrices of order three and |A| = -2, |B| = 5 and |C| = 3, ~~) then find: ¥ @ “241- Ql A\ = SxC-2<-16 — @ sari Pei”- (Sia) - (Pood) = Sree ren enecn tener eeeensecesacctenenescees ee eee aT aN : a oO Det. (2A*3B*4C)= | OA 3B 4e| = lx hec| = (a4) [allelic = (QYY C2) 5x3 Made with Xodo PDF Reader and Editor Trace of a Matrix 1 0 0 The sum of the Diagonal elements of a square matrix A is called the trace of A and is LetA=|0 a _ Bland |2A|? = 221 where «, 8 € Z, Then a value of a is: denoted by (t,.(A)). — ces chal b c ad ( SS) (in!) = L («+ P)(L-P)= 8x2 fA = |e then (t,(A)) =_O+ €+ l . 3 is hNiJ t¥(m) = Atd > a lal= at <+ = Q ( 5 4 KF = 2 la b Ow la\= ae : fA=|c fen 09) Net Ov ty : K=S Oo beer JA\= a2 B= 16 Made with Xodo PDF Reader and Editor Made with Xodo PDF Reader and Editor Vp N Vp N 1 1'420%+ 30 = 0 < 6 C1-L)CI-P) = 2 _ [125 * a = [40] (n-«) (x-®) = H 420x430 al = V, = . AG— 5 vale , Put He | then find the value of (1 — a)(1 — 8). ( \C ) =] =ft oO I-A) (|-B) = 1420430 = [° ae A v% [ ' -| Rg (3 ; () — i ] VV Made with Xodo PDF Reader and Editor Made with Xodo PDF Reader and Editor The Transpose of a Matrix @ QUESTION [JEE Adv. 2020] @) [Ans. 05] The trace of a square matrix is defined to be the sum of its diagonal entries. If A is a The matrix obtained by interchanging all the rows and columns is called transpose 2 x 2 matrix such that the trace of A is 3 and the trace of A° is -18, then the value of of a matrix. the determinant of A is ——————_ If Ais a square matrix, then |A| = |A‘|. cA @ ty(Ar)= -1% Made with Xodo PDF Reader and Editor Properties of Transpose If A’ and B' denote the transpose of A and B, (A + B)' = A’ +B"; here A and B must have the same order. AB)? = BTAT Sil A) : nee Gr > = Ai An ye (A ac) - CBA Made with Xodo PDF Reader and Editor QUESTION [JEE Mains 2024 (01 Feb. Shift-1) ] Dik2.,2\\)5 011 0 - all 1 o Layee Bd both side lat both sides 243 Ic\= JaliB}ar| Ixl= \at cal @ ~ = [arte a = @Yx\ 3) i Ic\ = 9 (KA)’ = KA*)k isa scalar (A”)T = (AT)" Made with Xodo PDF Reader and Editor pos, @ nt and Q = PAP’, then Pen Q= pap Pap’ QUESTION For the Matrices A = | = and B= E | , verify that (AB)' = B' A? Let A=[6 j] ana P= Ls: AB—@s)" SS some Ris. Bla at_, plat ©0000 @ [Ans. B] ,C = ABA! and X = A'C?A, then det X is equal to: \xl=? = [aC |\AI = Val icf yal = YAP yy = BY er = @)" Made with Xodo PDF Reader and Editor Tt “eye UE COS — sin 6 Tt sue HE —Ssin- Cos— 6 6 ell 1 2026 | 2. 2 ae Q= PAP o 2026) aay, Q”= PApt lod 2026_ Qos PA | 0 2026 202 r 0 2026 GT CA. @ [Ans. A] Le,9 Sir i -Sro G8 G&G, @ — Sin Sin 0 Pree f pas 2026 \) Made with Xodo PDF Reader and Editor Made with Xodo PDF Reader and Editor @ @ [Ans. B] HW QUESTION [JEE Mains 2025 (April) ] 7) Let a bea solution of x? +x +1 = oXand for some a and b in R, 1 16 2) be _ 4 m,n _ [4 a s- —-1 2/]=[0 0 Lata tS 3, then —2 -14 -8 For Example: p 4,7 343 a m + nis equal to: A- A= P 4 Uh "Af tin Cube Rovts of Vredey * Symmetric and Skew Symmetric Matrices Symmetric Matrix: A square matrix A = |a;,;| is said to be, symmetric if, aj; = aj, Vi and j, For symmetric matrix A = At. Made with Xodo PDF Reader and Editor Made with Xodo PDF Reader and Editor @ @ N=-p J}KAl= K"/A| Skew Symmetric Matrix: Square matrix A = [a:;| is said to be skew symmetric det both sidu 7 if aj; = —aj Vi and j. For a skew symmetric matrix A? = -A. . 7 _ ; For Example: za 0 -5 Aij=-Aju ¥ Ls | lA |= |-A] =. = = uk tz) Sf ° ks 2° mH 7 (Al= C\) [Al =5 {0 tc =- Aiv 1 —s 20ac~ =0 ic odd. ‘i | 3 — =) Cyeaw (Ala -\Al S$ -l0 3 AG chmet. —_—_—__-"- -——> z 1 ‘l 2 and ; : ja F If the sum of the diagonal elements of A is s, then _ is equal to + 2 214 4 -2C = | aot coe Alte 2, | *. 4 = 4A 4) > 2tl=3 [Ans. 5] Made with Xodo PDF Reader and Editor @ @ [Ans. B] QUESTION [JEE Mains 2023] The number of symmetric matrices of order 3, with all the entries from the set {0, 1,2,3,4,5, 6,7, 8, 9} is: 610 aA Io Ways oe b> |o Wyo Or Gar 3) 910 en A— Made with Xodo PDF Reader and Editor @ @ Properties of Symmetric and Skew Symmetric Matrices Ped Ais symmetric if A’ = A and A is skew symmetric if A’ = —A _ a b\w T fae Pils —> Symm A= ae a a AAT» SkKoW ex => Let A be any square matrix then| (A + A‘), AAT, ATA are all symmetric matrices whereas (A = AT) is always a skew symmetric matrix. T lt Ca A-@ Lot B — To frre B= 8 ae at ery? Harn Proved ap CA-AT) clu.ces3 SKay . Made with Xodo PDF Reader and Editor Made with Xodo PDF Reader and Editor Properties of Symmetric and Skew Symmetric Matrices Pb: If A and B are symmetric matrices then, At Cr Pere (a) AB + BAis asymmetric matrix. Cla (8) "+ G aT (b)} AB — BA isa skew symmetric matrix. gv ra x et pt (bo Lt D= AB- BF BaA+AB Ss Gah Ay" = glial pl gl > Apert = Made with Xodo PDF Reader and Editor QUESTION [JEE Mains 2023] Let A, B, C be 3 x 3 matrices such that A is symmetric and B and C (S1) Al’ B26 - B2°A?3 is symmetric — (s2)(A29c13 - CA? is symmetric Then, —_ - @ [Ans. A] QUESTION [JEE Adv. 2015] [Ans. C, D] > Zivsgy Symm Made with Xodo PDF Reader and Editor @ Let X and Y be two arbitrary, 3 x 3, non-zero, skew-symmetric matrices and Z be an arbitrary 3 x 3, non-zero, symmetric matrix. Then which of the following matrices is (are) skew symmetric? ——— KY 3 Ske 0. Y3Z4 - Z4Y3 ZA Sy ram % a Ly symm Ce) pie ie QO S¥men Symnp AB-®@fF ah Ayre 4 2 ea at 0 © 3) ey osu (8) Made with Xodo PDF Reader and Editor Mid Lecture Recap QUESTION True or False? a) If A&B are two square matrices then (A + B)* = (A? + B42 + 2AB) = q (b) ¢t,(AB) =t,(BA)) (T) (c) If a matrix A is multiplied by a non-zero Scalar “k’, then each element of the matrix gets multiplied with “k”. CT) (d) If order of A is (m x n) and order of B is (n x p), then order of AB is (n x p) (Fe) ; —_ eed (e) Asymmetric matrix is always a square matrix. (T) a en med ee (f) The determinant of askew symmetric matrix of odd order is always zero. (T) (g) If Ais any skew Symmetric matrix, then |A| is always zero. —— Made with Xodo PDF Reader and Editor Adjoint of a (2 « 2) Matrix by Trick CF) i) @ Made with Xodo PDF Reader and Editor @ Adjoint of a Matrix Let A = [a,] be a square matrix of order n and let C; be cofactor of a; in A, then the adjoint of A. denoted of A, denoted by adj A, is defined as the transpose of the cofactor matrix. T Cy, Cy2 Ci3 adj A=[C;]" > |C23 Caz C3 —— C3; C32 C33 Made with Xodo PDF Reader and Editor QUESTION [JEE Mains 2017] [Ans. C] IfA = ie ei then adj(3A?2 + 124A) is equal to: UZ 79 63 Ar 2 ey * cms 684 51 —A \i-“ tJ) \-t2 1% oF -72 94) 3A ( *® | — 63 51 —36 249 51 63] ane (24-26 84 72 YQ 2 51 84 2 = 63 72 SAH 2 pe les Ss aT Ody (aerian)- [S! 2 | 84 32 @ ©0000 
