A system of linear equations in three variables will be according to one of the following cases.

**Points to Remember**

·
A system of linear equations in three variables will be according
to one of the** **following cases.

(i) Unique solution

(ii) Infinitely many solutions

(iii) No solution

·
The least common multiple of two or more algebraic expressions is
the expression** **of lowest degree (or
power) such that the expressions exactly divides it.

·
A polynomial of degree two in variable** ***x*** **is called a quadratic
polynomial in** ***x*. Every** **quadratic polynomial can
have atmost two zeroes. Also the zeroes of a quadratic polynomial intersects
the *x*-axis

·
The roots of
the quadratic equation *ax* ^{2} + *bx* +*c*
= 0 , (*a* ≠ 0) are
given by

·
For a quadratic equation *ax* ^{2} + *bx* +*c*
≠ 0

·
If the roots of a quadratic equation are** ***α*** **and** ***β*** **, then the equation is
given by** ***x *^{2}* *−* *(*α *+*
β*)*x *+* αβ *=0* *.

·
The value of the discriminant** **( Δ = *b* ^{2} − 4*ac*) decides the nature
of roots as follows

(i) When Δ > 0 , the roots are real and unequal.

(ii) When Δ = 0 , the roots are real and equal.

(iii) When Δ < 0 , there are no real roots.

·
Solving quadratic equation graphically.

·
A matrix is a rectangular array of elements arranged in rows and columns.

·
Order of a matrix

If a matrix *A* has *m* number of rows and *n*
number of columns, then the order of the matrix *A* is (Number of rows)×(Number
of columns) that is, *m* ×*n* .We read *m* ×*n* as *m*
cross *n* or *m* by *n*. It may be noted that *m* ×*n*
is not a product of *m* and *n*.

·
Types of matrices

(i) A matrix is said to be a row matrix if it has only one row and any number of
columns. A row
matrix is also called as a row vector.

(ii)
A matrix is said to be a column matrix if it has
only one column and any number of rows. It is also called as a column vector.

(iii) A matrix in which the number of rows is equal to the number of columns is called a square matrix.

(iv) A square matrix, all of whose elements, except those in the
leading diagonal are zero is called a diagonal matrix.

(v) A diagonal matrix in which all the leading diagonal elements
are same is called a scalar matrix.

(vi) A square matrix in which elements in the leading diagonal are
all “1” and rest are all zero is called an identity matrix (or) unit matrix.

(vii) A matrix is said to be a zero matrix or null matrix if all its elements are zero.

(viii) If *A* is a matrix, the matrix obtained by
interchanging the rows and columns of *A *is called its transpose and is
denoted by* AT *.

(ix) A square matrix in which all the entries above the leading
diagonal are zero is called a lower triangular matrix.

If all the entries below the leading diagonal are zero, then it is
called an upper triangular matrix.

(x) Two matrices *A* and *B* are said to be equal if and
only if they have the same order and each element of matrix *A* is equal
to the corresponding element of matrix *B*. That is, *a* * _{ij}*
=

·
The negative of a matrix** ***A _{m}*

·
**Addition and subtraction of matrices**

Two matrices can be added or subtracted if they have the same
order. To add or subtract two matrices, simply add or subtract the
corresponding elements.

·
**Multiplication of matrix by a scalar**

We can multiply the elements of the given matrix *A* by a
non-zero number *k* to obtain a new matrix *kA* whose elements are
multiplied by *k*. The matrix *kA* is called scalar multiplication of
*A*.

Thus if *A* = (*a _{ij}* )

Tags : Algebra | Mathematics Algebra | Mathematics

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