Matrices and Determinants- 2 | Mathematics for Competitive Exams PDF Download

Properties on Matrices

1. A + B = B + A [Commutative]
2. (A + B) + C = A + (B + C) [Associativity]
3. A + O = A = O + A [Existence of Identity]
4. -A + A = O = A + (-A) [Existence of Inverse]
5. A + B = A + C ⇒ B = C ⇒ B + A = C + A ⇒ B = C [Cancellation law]
6. The equation A + X = O has a unique solution in the set of all A (m × n) matrices
7. k(A + B) = kA + kB [k be any scalar]
8. (p + q)A = pA + qA [p, q are two scalars]
9. (pq)A = p(qA)
10. (-K)A = -(KA) = K(-A)
11. 1A = A, (-1)A = -A
12. -(A + B) = -A - B
    
13. If the product AB exists, then it is not necessary that BA Product also exist.
    
14. A(BC) = (AB)C [Associativity]
    
15. A(B + C) = AB + AC [distributive]
    
16. The equation AB = 0 does not necessarily imply that at least one of matrices A and B must be a zero matrix.
    
17. If AB = 0, then it does not necessarily imply that BA = 0
    
18. A_II = A = I_nA
    
19. A² - B² = (A - B)(A + B) is true only if matrices A and B commute
    
20. The product of two triangular matrices is itself triangular.
    
21. If a diagonal matrix is commutative with every matrix of the same order then it is necessary a scalar matrix.
    
22. If AB = A and BA = B then A and B are idempotent
    
23. If B is an idempotent matrix, then A = I - B is also idempotent and that AB = BA = 0
    
24. A is involuntary iff (I + A)(I - A) = 0
    
25. tr(kA) = k.tr(A)
    
26. tr(A + B) = tr(A) + tr(B)
    
27. tr(AB) = tr(BA)
    
28. (A^T)^T = A
29. (A + B)^T = A^T + B^T
    
30. (AB)^T = B^T A^T
31.  Let  denote the Complex conjugate of A then
    
32. A is a skew-symmetric then is diagonal elements are all zero
33. The diagonal elements of a skew-symmetric matrix all are zero
34. Every diagonal element of Hermitian matrix must be real
35. The diagonal elements of skew-Hermitian matrix must be purely imaginary numbers or zero
36. For any non-zero real k if A is symmetric (skew-symmetric) then kA is also symmetric (skew-symmetric)
37. If A is Hermitian Matrix then iA is skew Hermitian
38. If A is a skew-Hermitian matrix, iA is Hermitian
39. (AB)^T = B^TA^T
40. For real matrices A, B and C
    a. A^TA = 0 iff A = 0
    b. AB = 0 iff A^TAB = 0
    c. AB = AC iff A^TAB = A^TAC
41. (AB)^T = A^TB^T iff AB = BA
42. If A and B be diagonal matrices then AB is also diagonal and that AB = BA
43. Let A be square real matrix then
    a. A + A^T is symmetric, even if A is not symmetric
    b. AB is not necessarily symmetric if A and B are
    c. A^TBA is symmetric if B is symmetric but that the converse need not be true
44. Let A be a square real matrix
    a. A - A^T is skew-symmetric
    b. A can be decomposed into the sum of a symmetric and a skew-symmetric matrix

Properties on Determinants

1. The sum of the products of the elements of any row or column with the corresponding cofactors of the matrix is equal to the value of the determinant.
2. The sum of the products of the elements of any row and the cofactors of some other row is zero.
3. Let ( A = (a_ij) be square matrix of order n , and let C_ij be the cofactor of a_ij then
   
4. The determinant of a triangular matrix is the product of its diagonal elements
5. For any two square matrices of the same order, |AB| = |A||B|
6. The value of the determinant of a diagonal matrix is equal to the product of the elements lying along its principal diagonal
7. The value of the determinant of a unit matrix is always equal to 1
8. The value of the determinant of a triangular matrix (Upper or lower) is equal to the product of the elements lying along its principal diagonal
9. The value of a determinant does not change when rows and columns are interchanged. |A| = |A^T|
10. If any two rows (or two columns) of a determinant are interchanged, the value of the determinant is multiplied by -1
11. If all the elements of one row (or one column) of a determinant are multiplied by the same number k, the value of the row determinant is k times the value of the determinant.
12. If A be an n-rowed square matrix, and k be any scalar, then |kA| = kⁿ|A|
13. If two rows (or two columns) of a determinant are identical, the value of the determinant is zero
14. The value of the determinant of a skew-symmetric odd order is always zero
15. If in a determinant each element in any row or column consists of the sum of two terms, then the determinant can be expressed as the sum of two determinants of the same order
16. If to the element of a row (or column) of a determinant is added M times the corresponding elements of another row (or column), the value of the determinant thus obtained is equal to the value of the original determinant
17. Let A be a square matrix then
    a. |A^T| = |A|
    b. |A| = |A|
    c. |A^θ| = |A|

Properties on Inverse of a Matrix

1. (Adj A)A = A(Adj A) = |A|I_n
2. Every invertible matrix possess a unique inverse
3. (AB)^-1 = B^-1A^-1
4. (A^T)^-1 = (A^-1)^T
5. (A^-1)^θ = (A^θ)^-1
6. adj . 0 = 0
7. adj . I_n = I_n
8. adj A^T = (adj A)^T
9. If A is a symmetric matrix, then adj A is also symmetric
10. If A is symmetric then A^-1 is also symmetric
11. The adjoint of a diagonal matrix of order 3 is a diagonal matrix
12. 

    det(A^-1) = (det A)^-1

13. If B is non-singular, then matrices A and B^-1AB have the same det A & B of A
14. If matrices A and B commute, then A^-1 and B^-1 also commute
15. If A, B, C are three matrices conformable for multiplication then (ABC)^-1 = C^-1B^-1A^-1
16. If a matrix A is non-singular, then AB = AC ⇒ B = C, where B and C are square matrices of the same order as A
17. If the product of two non-zero square matrices is a zero matrix, then both of them must be singular.
18. If |A| = 0, then |adj A| = 0
19. adj(AB) = adj B · adj A
20. If A · B be n-rowed orthogonal matrices AB and BA are also orthogonal matrices
21. If A, B be n-rowed unitary matrices, AB and BA are also unitary
    

Rank of a Matrix

1. The rank of a matrix A is r if it possess the following two properties
   a. There is at least one square sub matrix of A of order r whose determinant is not equal to zero
   b. And all the matrix A Counting any square sub matrix of order r+1, then the determinant of every square sub matrix of A of order r+1 should be zero.
2. The rank of matrix is the order of any highest order non-vanishing minor of the matrix.
3. The rank of every non-singular matrix of order n is n.
4. The rank of a square matrix A of order n less than n iff A is singular i.e. |A| = 0
5. The rank of every non-zero matrix is ≥ 1
6. The rank of a matrix is ≤ r, if all (r+1)-rowed minors of the matrix vanish
7. The rank of a matrix is ≥ r, if there exists r-rowed minor of the matrix which is not equal to zero
8. The rank of a matrix in Echelon form is equal to the number of non-zero rows of the matrix.
9. The rank of the transpose of a matrix is same as that of original matrix.
10. The rank of matrix every element of which is unity is 1
11. A is a non-zero column and B a non-zero row matrix then rank(AB) = 1
12. Rank of a matrix is ≥ the rank of every sub matrix
13. Rank of skew-symmetric matrix cannot be rank 1
14. Let A be an m x n matrix.
    a. 0 ≤ rank(A) ≤ min{m,n}
    b. rank(A) = 0 ⇔ A = 0
    c. rank(Iₙ) = n
    d. rank(kA) = k.rank(A) if k ≠ 0
15. Let A be an m x n matrix and let Ax = 0 for some x ≠ 0 ⇔ rank(A) ≤ n - 1
16. The rank of diagonal matrix equals the number of non-zero diagonal elements it possess but is not true for triangular matrix.
17. Every m x n matrix A of rank r can be written as A = B Cᵀ, where B(m x r) and C(m x r) both have rank r
    rank(A + B) ≤ rank(A) + rank(B).
18. rank(A + B) ≤ rank(A) + rank(B)
19. rank(A - B) ≥ |rank(A) - rank(B)|
20. rank(AB) ≤ min(rank(A), rank(B))
21. rank(AB) ≠ rank(BA)
22. Let A₁ be a sub matrix of A then rank(A₁) ≤ rank(A)
23. rank(AᵀAB) = rank(AB) = rank(ABBᵀ) for any conformable matrices A and B.
24. rank(A) = rank(Aᵀ) = rank(AAᵀ) = rank(AᵀA)
25. Let A be an m × n matrix, and let B(m × n) and C(m × n) be non-singular
    a. rank(BA) = rank(A)
    b. rank(AC) = rank(A)
    c. rank(BAC) = rank(A)
26. For conformable matrices A, B, C
    a. rank(BA) = rank(BᵀBA)
    b. rank(AC) = rank(ACCᵀ)
    c. rank(BA) = rank(A)
    d. rank(AC) = rank(A)

Properties on Trace

1. Let A and B be square matrices of the same order, and let k and m be positive scalars, then
   a. tr(A + B) = tr(A) + tr(B)
   b. tr(kA) = ktr(A)
   c. tr(kA + mB) = ktr(A) + mtr(B)
   d. tr(A^T) = tr(A)
   
2. For any real matrix A, tr(A^TA) ≥ 0, with tr(A^TA) = 0 iff A = 0
3. Let A and B be m x n matrices, then tr(A^TB) = tr(BA^T) = tr(AB^T) = tr(B^TA)
4. tr(ABC) = tr(CAB) = tr(BCA) but tr(ABC) = tr(ACB) is not true
5. A real square matrix A is normal if A^TA = AA^T
   a. Every symmetric matrix is normal.
   b. Every orthogonal matrix is normal
   c. Let A be a normal 2 x 2 matrix then A is either symmetric or has the form 
   
6. A^P is symmetric when A is symmetric.
7. Consider an n×n triangular matrix A. Then all the positive integral power of A are also triangular.
8. For two real matrices A and B of the same order then Inner product is defined as ⟨A,B⟩=∑_i,j a_ijb_ij=tr A^TB
   a. ⟨A,B⟩=⟨B,A⟩
   b. ⟨A,B+C⟩=⟨A,B⟩+⟨A,C⟩
   c. ⟨kA,B⟩=k⟨A,B⟩
   d. ⟨A,A⟩≥0 with ⟨A,A⟩=0⇔A=0
9. For a real matrix A, the Norm of a matrix is defined as
   a. ‖kA‖=|k|.‖A‖b. ‖A‖≥0, with ‖A‖=0 iff A=0
   c. ‖A+B‖≤‖A‖+‖B‖ (triangular inequality)
10. Let A and B are square matrix of same of order. AB = I if BA = I
11. Let A be an orthogonal matrix of order n
    a. A is non-singular
    b. A^T = A^-1
    c. A^T is orthogonal
    d. AB is orthogonal when B is orthogonal
12. If A is orthogonal, |A| = ±1
13. If A is non-singular, |A^-1| = |A|^-1
14. Let  then
    

15. 

16. If A and D are square, not necessarily of the same order,

17. where A and D are square matrices, not necessarily of the same order.
    a. If Z is symmetric then A and D are symmetric.
    b. If Z is diagonal then A and D are diagonal.
    c. If Z is upper triangular then A and D are upper triangular.
    d. All the above three statements are necessary but not sufficient.