If we have two linear equations
a11 y1 + a12 y2 =X1
a21 y1 + a22 y2 = X2
|a11 a12|
|a21. a22| is a determinant
Determinant is a number associated with every square matrix. The number which can tell you about whole matrix.
E.g. Matrix is singular when , |A|=0.
Matrix is inversible when |A|≠ 0.
This number packs in much information as possible.
Properties of determinants
1. |I|=1
2. If two rows are equal, then det =0
3. Exchanging the rows of det,reverse the sign of det.
4. Det behaves like a linear functions.
|a+a'. b+b'|. | a. b| + |a'. b'|
. |c. d. | = | c. d|. |c. d|
5. |ta. tb| . |a. b|
. |c. d |= t|c. d|
6. detAB= (detA) (detB)
7. detA^2=(detA)^2
8. det2A ≠2 detA =2^n (detA)
. nxn matrix
9. Determinant of transpose is equal to determinant
System of linear equations are solved by forming tables of coefficient's is called Matrix.
Matrix is every where, even in this page of my blog. In surface of velocities of occean, our own physical body has matrix within us. In blood ,blood clotting, matrix prevails which helps through computer simulation in bypass. Search engines use matrix to find which word occur in which web site.
Search engine creates a table,each row corresponds to each word searched, each colum corresponds to web pages, where the particular word found.
If the key word found successfully, the table has 1 corresponds to web page, and 0 if not found.
Matrix are supposed to be in our brain too.
Matrix can be favourite, like upper triangle and lower triangle matrix.
The high school linear algebra, the system of linear equations is omnipresent.
Multiplication of a matrix by a scalar
|KA|= K^n|A| n is the order of matrix
Condition required for two matrix to multiply.
A=[aij] mxn. matrix and B=[bij] nxp matrix
So, matrix A and B can. multiply only if number of columns in A = number of Rows in B
i.e C=AxB = [Cij] mxp
Properties of matrix multiplication
# Matrix multiplication is not commutative
AB≠BA.
If A is 2x4 matrix and B is 4x1 matrix then
AB is possible i.e multiplication is possible.
In case of BA to happen, its not possible because number of columns of B not equal to number of rows of A
Hence AB≠BA
#If AB= BA then matrices A and B are commutative
Now AI=IA=A
Hence, identity matrix is commutative with any square matrix.
# If AB = -BA then A and B are anti commutative.
If AB = BA then they're commutative
If AB = -BA then they're anti commutative
# If A and B are commutative matrices then
(A+B)^2 = A^2 +2AB +B^2
(A+B)^2 = (A+B) (A+B)
. = A^2+ AB+BA+B^2
Now, AB=BA only when AB are commutative matrix.
Therefore, (A+B)^2=A^2+AB+AB+B^2
. = A^2+2AB+B^2.
If A and B are commutative, then we can write
(A+B)^3 = A^3+3A^2 B+3AB^2+B^3
# Matrix multiplication is associative
A(BC)=(AB)C
# If A.B=A.C this does not imply B=C even if A is not a null matrix
# If A and Bare square matrix of same order then |AB|=|BA|= |A| |B| even though AB≠BA
Orthogonal matrix
A. Transpose of A = I then A is an orthogonal matrix
Each row and column of orthogonal matrix can be treayed as a normalised vector.
Any two rows or any two columns of orthogonal matrix can be treated as a orthogonal vectors.
Determinant of orthogonal matrix is +1 or-1
AA^T =I
det(A) det(A^T) =1
(a) (a)=1
a^2=1 therefore a=+-1
Unitary Matrix
AA^T=A^TA=I
A is a unitary matrix
Properties of Unitary matrix
1) Determinant of unitary matrix will be of unit modulus
2) Each row and column of unitary matrix can be treated as Normalised vector.
3) Any two rows or columns can be treated as Orthogonal vectrors.
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