Rank of a Matrix and its Properties PDF

Rank of a Matrix and Its Properties Very often, in Linear Algebra, you will be asked to find the rank of a matrix. You will need to solve problems based on the properties of the rank of a matrix. Useful for all streams of GATE (EC, EE, ME, CE, CS etc.) as well as IES and other competitive exams lik...

Rank of a Matrix and Its Properties Very often, in Linear Algebra, you will be asked to find the rank of a matrix. You will need to solve problems based on the properties of the rank of a matrix. Useful for all streams of GATE (EC, EE, ME, CE, CS etc.) as well as IES and other competitive exams like BSNL, DRDO, BARC etc. If you haven’t already read it, please check the previous articles on Types of Matrices and Properties of Matrices, to giveyourself a solid foundation before proceeding to this article. 𝐑𝐚𝐧𝐤 𝐨𝐟 𝐚 𝐌𝐚𝐭𝐫𝐢𝐱: The order of highest order non − zero minor is said to be the rank of a matrix. That means, the rank of a matrix is ‘r’ if i. All the minors of order (r + 1) and more if exists, are should be zero. ii. There exists at least one non − zero minor of order ‘r’. 𝐀𝐧𝐨𝐭𝐡𝐞𝐫 𝐃𝐞𝐟𝐢𝐧𝐢𝐭𝐢𝐨𝐧: The number of Non − zero rows present in the Matrix Echelon form is also known as Rank of a matrix 1|P age 𝐏𝐫𝐨𝐩𝐞𝐫𝐭𝐢𝐞𝐬 𝐨𝐟 𝐚 𝐑𝐚𝐧𝐤 𝐨𝐟 𝐚 𝐌𝐚𝐭𝐫𝐢𝐱: i. Only Rank of Null Matrix is zero ii. ρ(In) = n where In = unit matrix of order n iii. Rank of a matrix Am×n , ρ(Am×n) ≤ Min(m, n) iv. ρ(An×n) = n if |A| ≠ 0< n if |A| = 0 v. If ρ(A) = m and ρ(B) = n then ρ(AB) ≤ min(m, n) vi. If A and B are square matrices of same order ‘n’ then ρ(AB) ≥ ρ(A) + ρ(B) – n vii. If Am × 1 is a non zero column matrix and B1 × n is a non − zero row matrix then ρ(AB) = 1 viii. If A−1 is exists and B is a matrix of any order then ρ(AB)does not dependent on the matrix A i. e. ρ(AB) = ρ(B) ix. ρ(A + B) ≤ ρ(A) + ρ(B) x. ρ(A − B) ≥ ρ(A) – ρ(B) xi. ρ(AT) = ρ(A) 2|P age 𝐄𝐜𝐡𝐞𝐥𝐨𝐧 𝐅𝐨𝐫𝐦 𝐨𝐟 𝐚 𝐌𝐚𝐭𝐫𝐢𝐱: A matrix is said to be in echelon form if i. There exists any zero row, they should be placed below the non − zero row. ii. Number of zeros before a non − zero element is a row should increase according with row number. 1 2 3 → 𝐄𝐱𝐚𝐦𝐩𝐥𝐞: 1) A = 0 3 2 ⇒ ρ(A) = 3 = Number of Non zero rows 0 0 1 → 𝐍𝐨𝐭𝐞: i. To reduce a matrix into its echelon form only elementary row transformations are applied ii. Nullity of a matrix n − r. where n = order of a matrix and r = rank of a matrix iii. The Rank of a non − zero Skew symmetric of order not equal to zero at any time. iv. The example given below explains the procedure to calculate rank of a matrix in two methods i. e. in normal method and Echelon form Method. 3|P age → 𝐄𝐱𝐚𝐦𝐩𝐥𝐞: 1 2 1 Find the Rank and Nullity of the matrix A = −2 −3 1 3 5 0 4|P age 𝐌𝐞𝐭𝐡𝐨𝐝 𝟏: 𝐍𝐨𝐫𝐦𝐚𝐥 𝐦𝐞𝐭𝐡𝐨𝐝 1 2 1 Given A = −2 −3 1 and order n = 3 3 5 0 |A| = 1(0 − 5) − 2(0 − 3) + 1(−10 + 9) = −5 + 6 − 1 = 0 i. e. |A| = 0 ⇒ rank of matrix is less than order = 3 i. e. ρ(A) < 3 1 2 Consider | | = (−3) + 4 = 1 ≠ 0 −2 −3 ∴ The rank of a matrix, r = 2 (∵ high order of non zero minor order) Nullity of a matrix = n – r = 3 – 2 = 1 𝐌𝐞𝐭𝐡𝐨𝐝 𝟐: 𝐄𝐜𝐡𝐞𝐥𝐨𝐧 𝐅𝐨𝐫𝐦 𝐌𝐞𝐭𝐡𝐨𝐝 1 2 1 A = [−2 −3 1] 3 5 0 R2 ∶ R2 + 2R1 R3 ∶ R3 − 3R1 1 2 1 ⇒ A = [0 1 3] 0 −1 −3 R3 ∶ R3 + R2 1 2 1 ∴ A = [0 1 3] ` 0 0 0 ∴ Rank of a matrix = r = 2 (Number of non − zero rows) 5|P age Nullity of a matrix, n – r = 3 – 2 = 1 → 𝐓𝐈𝐏: 1) If order of a matrix is ≤ 3. Then apply Normal Method to find the rank of a matrix. 2) If the order of matrix is > 3 , then apply Echelon method to find the rank of a matrix. 6|P age