Good morning students. Today we will talk about topic, Matrices and Determinants. We

know what is Matrix it is a rectangular arrangement of numbers and what is Determinant, it

is a square arrangement of numbers. Determinant is always associated with a value. So we

know many things about Matrices and Determinants, and we know that there are many

properties of both Matrices and Determinants. For example, we know that in Determinants,

if two rows or two columns are identical then value of Determinant is zero, if we interchange

two rows then value of Determinant gets multiplied by a negative sign. If we interchange

rows and columns then the value of Determinant remains same. We also know we can

apply operations like ith row goes to ith row minus some constant times and jth row. And by

doing this value of Determinant does not get changed. And there are many other properties

of Matrices and Determinants. Let us talk about few questions which can come in the

competition examinations. We will give you a quick glimpse of such type of questions.

First question of your worksheet, if a is greater than zero and discriminant of this quadratic is

negative then this Determinant is what? a is greater than zero and discriminant is negative

that is four times b square minus ac is less than zero. We also know if a is greater than zero

and discriminant is negative then this quadratic is greater than zero for all x. Then this

quadratic is always greater than zero. This is what we know from quadratic equations.

Let us talk about Determinant a, b, ax plus b, b, c, bx plus c, ax plus b, bx plus c, zero. By

applying R3 goes to R3 minus R1 into x minus R2. From row three multiply row one and

subtract it and also subtract row two, what do we get is, first two rows will remain same, this

will become zero, bx plus c minus bx plus c will again become zero and then what does this

become, ax square plus bx plus bx plus c that is ax square plus 2bx plus c with the negative

sign.

Now opening using the third row, if we expand from the third row, we will get value of

Determinant is equal to minus times ax square plus two bx plus c into ac minus b square.

That is, b square minus ac into ax square plus bx plus a square plus 2 bx plus c. Now let us

see what is the correct answer, we have opened this Determinant and it is this, this

Determinant is not equal to option b. Option b is incorrect because it is negative sign.

This zero is also wrong. Now is this positive or negative? b square minus ac is negative, and

ax square plus 2bx plus c is positive, negative into positive is negative, hence Determinant is

always negative. Answer is option c. Simply applying one property and after expanding it

we got the value, after that we saw that the sign was negative, and we got the answer.

Questions are generally based on simple properties of determinants. Let us move forward.

Now see the second question on your worksheet.

If l, m, n are pth, qth and rth term of GP, let us assume l is equal to A into K raised to the

power P minus one, m is equal to A into K raised to the power q minus one, n is equal to A

into K raised to the power r minus one.

GP where the first term is A and common ratio r, so write the values of l, m, n they are pth,

qth and rth term of GP. We have to talk about the value of the Determinant which is equal

to log l p 1, log m q 1, log n r 1. Let us put value of log l, what is log l? Log A into K raised to

power of p minus 1, log A into K raised to the power q minus 1, log A into K raised

to the power r minus one. This is value of Determinant, let us simplify this, log A plus p

minus 1 log k, value of Determinant becomes this. Now let us apply C one goes to C one

minus log A into C 3, column first goes to column first minus log A into column third.

Applying this value of.