Determinants and Matrices
Determinants
 A square array or arrangement of terms called elements or entries with m columns
and n rows (m = n) enclosed by vertical bars and which represents a certain value.
 It is used in the analysis of linear equations and their solution. (Cramer’s Rule)
Important Properties of Determinants
(v) The value of the determinant D is equal to the summation of the product of the
elements of a particular row or column with their respective cofactors
Cramer’s Rule
 Utilizes determinants in finding the values of the variables in the system of linear
equation.
Matrices
 A matrix is a rectangular arrangement of numbers into rows and columns.
For example, matrix A has two rows and three columns
Matrix Dimensions - tells its size: the number of rows and columns of the matrix, in that
order.eg 2X3, 3X2
Matrix elements - is simply a matrix entry. Each element in a matrix is identified by naming the
row and column in which it appears
In general, the element in row I and column j of matrix A is denoted as aij
Matrix Operation
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Addition of Matrices
Subtraction of Matrices
Scalar Multiplication
Multiplication of Matrices
Transpose of Matrices
The addition of matrices can only be possible if the number of rows and columns of both the
matrices are the same. While adding 2 matrices, we add the corresponding elements
Matrices subtraction is also possible only if the number of rows and columns of both the
matrices are the same. While subtracting 2 matrices, we subtract the corresponding elements.
Scalar Multiplication
The product of a matrix A with any number 'c' is obtained by multiplying every entry of the matrix
A by c, is called scalar multiplication. i.e., (cA)ijij = c(Aijij)
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,
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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
Multiplication of Matrices
Matrices multiplication is defined only if the number of columns in the first matrix and
rows in the second matrix are equal.
Example:
Show that:
Special Matrices:
1. Transpose of a Matrix
2. Symmetric Matrix
3. Skew Symmetric
4. Singular and Non-singular Matrices
5. Adjoint of a Matrix
6. Inverse of a Matrix
Group Activity 1:
Solve the following problems manually:
1. Use matrices to find the solution set of the following sets of linear equations:
a.
b.
c.
2.
Find :
a. AB
b. BC
c. ABC
3. Find the inverse of the coefficient matrices in problem 1.