Matrices

Matrices is a plural form of a matrix, matrix is the arrangement of numbers in  the rectangular form within the square brackets.

👉The numbers or entries in the matrix are known as its elements. Horizontal entries of matrices are called rows and vertical entries are known as columns.

Order of the matrix

👉If a matrix has m rows and n columns, then it will have m × n elements. A matrix is represented by the uppercase letter, in this case, 'A', and the elements in the matrix are represented by the lower case letter.

Types of Matrices

There are various types of matrices based on the number of elements and the arrangement of elements in them.

Row matrix: A row matrix is a matrix having a single row is called a row matrix. 

Column matrix: A column matrix is a matrix having a single column is called a column matrix. 

Square matrix: A matrix having equal number of rows and columns is called a square matrix. 

Rectangular Matrix: A matrix having unequal number of rows and columns is called a rectangular matrix. 

Diagonal matrices: A matrix with all non-diagonal elements to be zeros is known as a diagonal matrix.

Scalar matrix: 
Identity matrices: A diagonal matrix having all the diagonal elements equal to 1 is called an identity matrix.

Operations on the Matrices:

The basic operations that can be performed on matrices are:

• Addition of Matrices
• Subtraction of Matrices
• Scalar Multiplication
• Multiplication of Matrices
• Transpose of Matrices

 Properties of scalar multiplication in matrices

The different properties of matrices for scalar multiplication of any scalars K and l, with matrices A and B are given as,

• K(A + B) = KA + KB
• (K + l)A = KA + lA
• (Kl)A = K(lA) = l(KA)
• (-K)A = -(KA) = K(-A)
• 1·A = A
• (-1)A = -A

Properties of Matrix Multiplication

There are different properties associated with the multiplication of matrices. For any three matrices A, B, and C:

• AB ≠ BA
• A(BC) = (AB)C
• A(B + C) = AB + AC
• (A + B)C = AC + BC
• A$I_m$ = A = AIn, for identity matrices I${}_m$ and In.
• A${}_{m\times n}$O${}_{n\times p}$ = O${}_{m\times p}$, where O is a null matrix.

Transpose of Matrix

Properties of transposition in matrices

There are various properties associated with transposition. For matrices A and B, given as,

• (A^T)^T = A
• (A + B)^T = A^T + B^T, A and B being of the same order.
• (KA)^T= KA^T, K is any scalar(real or complex).
• (AB)^T= B^TA^T, A and B being conformable for the product AB. (This is also called reversal law.)

Symmetric and skew-symmetric matrices:

Symmetric matrices: A square matrix D of size n×n is considered to be symmetric if and only if A^T= A. 

Skew-symmetric matrices-A square matrix F of size n×n is considered to be skew-symmetric if and only if A^T= - A.

Matrices Formulas

There are different formulas associated with matrix operations depending upon the type of matrix. Some of the matrices formulas are listed below:

• A(adj A) = (adj A) A = | A | In
• | adj A | = | A |^n-1
• adj (adj A) = | A |^n-2 A
• | adj (adj A) | = | A |^{(n-1)^2}
• adj (AB) = (adj B) (adj A)
• adj (A^m) = (adj A)^m,
• adj (kA) = k^n-1 (adj A) , k ∈ R
• adj(In) = In
• adj 0 = 0
• A^-1 = (1/|A|) adj A
• (AB)^-1 = B^-1A^-1

Most Important Properties

Addition and Subtraction of matrices

A + B = B + A

(A + B) + C= A + (B + C)

k (A + B)=kA + kB

Multiplication of Matrices

AB ≠ BA

(AB)C =A(BC)

Distributive law

A (B + C) = AB + AC

(A + B) C =AC+BC

Multiplicative identity for a square Matrix A;   AI = IA = A

 Properties of Transpose of Matrix

(A^T)^T   = A

(k A)^T   = k A^T

(A+B)^T   =   A^T + B^T

(A B)^T   =   B^T   A^T

Symmetric and Skew Symmetric Matrices

Symmetric Matrix –if A^T=A

Skew Symmetric Matrix– If A^TA

Note: In Skew matrix diagonal  element are always 0

For any square matrix A,

(A+A^T) is a symmetric matrix

(A–A^T) is a skew symmetric matrix

Properties of inverse

1. For matrix A, A^–1 is unique, i.e. there is only one inverse of a matrix

2. (A^–1)^–1 = A

3. (kA) = 1/k  A^–1

4.(A^–1)^T =(A^T)^–1

5. (A + B)^–1 =A^–1 + B^–1

6. (AB)^–1 = B^–1A^–1

Properties of Determinants

1. Determinant of any identity matrix is 1   or  | A | = 1

2. I A^TI =I A I

3. IABI =IAI IBI

4. IA^–1I =1/ | A |

5. IkAI = kⁿ | A |  where n is order of the matrix

6. Similarly I –A I = I –1× A I

                          =(–1)ⁿ × IAI

7. (adj A) A =A (adj A) = IAI I

8. Determinant of adj A;

                        Iadj AI = | A |^n–1 ; where n is order of the matrix

 9. adj (adj A) = | A |^n-2 A

10. | adj (adj A) | = | A |^{(n-1)^2}

11. adj (AB) = (adj B) (adj A)

12. adj (A^m) = (adj A)^m,

13. adj (kA) = k^n-1 (adj A) , k ∈ R

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Math Symbol	Description	Alt-Code
+	plus	Alt-43
-	minus	Alt-45 or Alt-8722
×	times or multiply	Alt-0215
÷	division or divided by	Alt-0247
≠	not equal to	Alt-8800
<	less than	Alt-60
>	greater than	Alt-62
≤	less than or equal to	Alt-243 or Alt-8804
≥	greater than or equal to	Alt-242 or Alt-8805
±	plus or minus	Alt-241 or Alt-0177
≈	approximately equal to	Alt-247
°	degrees	Alt-0176
µ	micro	Alt-230
√	square root	Alt-251 or Alt-8730
²	superscript 2 or squared	Alt-253 or Alt-0178
³	superscript 3 or cubed	Alt-0179
π	pi	Alt-0227
∞	infinity	Alt-236 or Alt-8734
▲	triangle	Alt-30
~	similar to or tilde	Alt-126
┬	perpendicular with	Alt-10178
|	vertical line or bar	Alt-124
∫	integral	Alt-8747
→	right arrow	Alt-26
∟	right angle	Alt-28 or Alt-8735
¼	one-fourth	Alt-172 or Alt-0188
½	one-half	Alt-171 or Alt-0189
¾	three-fourths	Alt-190
⅓	one-third	Alt-8531
⅔	two-thirds	Alt-8532