KCET MATRICES AND DETERMINANTS Synopsis Theory

Added: 2026-06-01

[00:00:01]Hello dear students.

[00:00:03]In this video, we are beginning with that is the KCET class on two topics, matrices and determinants.

[00:00:11]Matrices and determinants both are connected to each other.

[00:00:15]We know determinants are defined students by taking that is a square matrix.

[00:00:20]That's why these two chapters are connected. Hence, we discuss that is KCET examples on both the topics at a time.

[00:00:30]When we see regarding the weightage of this chapter, every year students, they are asking minimum six questions on these two that is matrices and determinants and maximum that is eight questions. If I see the last year, that is 2025 KCET question paper, they have asked students eight questions on these two topics. So, remember weightage for the chapter is matrices and determinants approximately that is six to eight marks. Minimum six marks, maximum eight marks you you may get for these two topics that is matrices and determinants. So, in today's class, we are discussing that is the synopsis. I mean, what you learned in the theory uh theory class for matrices and determinants, these are just the meanings we have some terms meanings.

[00:01:25]So, I'll just read out so you just recall all these things which you learned already the theory. So, here just we are revising. So, first we discuss students that is what we learned in the theory. And next, we'll see what are the shortcuts we can use for these KCET examples for these two chapters that is matrices and determinants. So, let us begin with the synopsis. A set of m into n numbers or functions. So, whenever they give that is numbers or functions, if you arrange all those numbers or functions in m rows and n columns, then that is what we call it as a matrix and its order is taken as students m cross n.

[00:02:06]In the matrices, we have a different different types. That is a number one, row matrix. A matrix which has exactly one row is called as a row matrix. A matrix which has exactly one column is called as a column matrix. Then comes zero matrix which is also called as null matrix. A matrix in which all the elements are zeros. Then that is what we call it as a zero matrix and it is always denoted by that is capital letter O. Rectangular matrix means it looks like a rectangle. So, then definitely students, the number of rows and the number of columns are not same, then that matrix looks like a rectangle and hence it is called as a rectangular matrix. Square matrix means the number of rows and the number of columns are that is same. So, whenever you have a m cross n matrix, then this is said to be square matrix only when that is m is equal to n. Next comes that is diagonal matrix. A square matrix in which all the non-diagonal elements must be zero. So, whenever the non-diagonal elements are zero, the diagonal is highlighted like and hence students, it is called as that is diagonal matrix. So, you remember for all i not equal to j. i not equal to j means these are the non-diagonal elements and all the non-diagonal values are how many here? That is zero.

[00:03:24]And thus we say that a square matrix in which all the non-diagonal elements are zeros, then it is called as diagonal matrix. Next comes that is scalar matrix. So, scalar matrix means first it should be a diagonal matrix and in that students, diagonal elements must be same.

[00:03:44]Non-diagonal elements are already zeros.

[00:03:46]Along with that, if the diagonal elements are also same, then that matrix is called as scalar matrix. Further, if I take a scalar matrix and in that again the diagonal elements are one, already non-diagonal elements are zeros. If the diagonal elements are also one, then that matrix is called as identity matrix or sometimes it is called as students unit matrix and it is denoted by always capital letter I. Thus we say the elements in the identity matrix are for all i is equal to j. I mean, diagonal elements are always one and the non-diagonal elements are always that is zero. This is regarding that is identity matrix. Next comes triangular matrix.

[00:04:26]So, in a square matrix, whatever the principal diagonal is there, above the diagonal or below the diagonal, if all elements are zero, then it is called as that is triangular matrix. In this we have a two types, upper and lower.

[00:04:41]Suppose [snorts] the in a square matrix that is in which all the elements below the diagonal if they are zero, suppose below you are getting the zeros, then triangle you will get in the above side, that is upper side.

[00:04:54]That's why that triangular matrix is called as upper triangular matrix. And suppose if you have that is above the diagonal all the elements are zero, then you will get below the triangular triangle shape. So, that is what they have called it as a lower triangular matrix. Reverse you should remember if the zeros are below, it's a upper triangular and if zeros are above, then it is a lower triangular matrix. This is regarding the types of matrices. Next comes algebra of matrices. That is number one, equality. Two matrices [snorts] are said to be equal whenever they have a same order and the corresponding elements are same. That is nothing but suppose if you have one uh matrix size aij of order m cross n and another matrix b that is having elements bij of order p cross q, then both the matrices are said to be equal students. Number one, the m must be same as p and n must be same as q. That is what we say both must have same order.

[00:05:52]And the second one, the corresponding elements means suppose if I take a11 from this, then b11 from this, both must be same. If I take a12, then b12, they must be same. So, for all i and j, if aij is equal to bij and along with that the order is also same, then we say that these two matrices are how? That is equal. Next comes matrix addition.

[00:06:17]Matrix addition and as well as matrix subtraction students, these are defined only for the matrices which are of same order. So, whenever you have a and b are same order, then only you will get a plus b. And that addition is defined like this. Whenever you have two matrices of order m cross n, then a plus b will be corresponding elements you have to add it. Means a11 from this, b11 from this you should add.

[00:06:45]a12 from this, b12 from this you should add. Like that you can continue and that matrix is also of order what you are getting, that is m cross n. This is regarding matrix addition. Along with this matrix addition students, you know matrix addition satisfy some of the uh laws, that is properties you can say.

[00:07:05]Number one, commutative law. a plus b is always equal to that is b plus a.

[00:07:11]Commutative law holds good for matrix addition. Next comes associative law.

[00:07:16]Matrix addition is associative. That is a plus in the bracket b plus c is always same as in the bracket a plus b plus c.

[00:07:24]Next, distributive law also holds good.

[00:07:27]Whenever you have a scalar into bracket a plus b that is matrix, so it can written as ka plus kb. Similarly, if you have k plus l, that is k is a scalar, l is a scalar into a if it is there, you can write this as a ka plus la.

[00:07:44]And one more you can say that is Suppose if you have k plus l into a plus b, so k plus l that is these are the scalars and a plus b, a and b are the matrices. Then this can written as k plus l into a plus k plus l into b. That is the third one. Next comes additive identity. As we know that is zero matrix is called as students additive identity.

[00:08:13]Whenever we add zero matrix to any matrix, the again answer is that matrix a only. That is what we say a plus o or o plus a, the answer is again a.

[00:08:22]Whenever this happens, the o is called as that is additive identity. So, remember zero matrix is additive identity. Next additive inverse in the sense whenever we add the matrix a to another matrix such that the addition is o, then that we should take it as a minus a. So, this minus a is called as additive inverse of the matrix a. So, here two things you should remember.

[00:08:46]Zero matrix is called as additive identity and minus a is called as additive inverse of the matrix a.

[00:08:53]Next comes matrix subtraction. So, matrix subtraction is same to same as that is matrix addition. Have you ever adding corresponding elements? No, here students, you have to do corresponding elements subtraction. And again point to be noted, matrix subtraction is defined only for the matrices which are of same order. That is what you should remember.

[00:09:14]And here, matrix subtraction doesn't satisfy any properties. Here no, commutative law, associative law, these doesn't holds good. Hence students, we don't have properties for this. So, we have properties only for the addition and as well as that is matrix multiplication. Okay. Matrix addition and matrix subtraction both are same.

[00:09:33]That is uh conditions are same to define a plus b and a minus b. But for matrix multiplication students, we have a different condition here. We are not doing here the corresponding elements multiplication, okay? Matrix multiplication is defined only when the whatever the first matrix is there in this columns and the second matrix rows, whenever these are same. First matrix columns and the second matrix rows are same. In that case students, matrix multiplication is defined. And how they do? Yes, that is row to column multiplication. Suppose I'm having here A is a matrix of order that is m cross p and B is a matrix of order p cross n.

[00:10:13]You can observe here the first matrix columns are p and the second matrix rows are also p. So here, matrix multiplication is defined. And whatever that AB we are getting now after multiplication, that is a matrix of order we will get, that is m cross n. So that is what you should remember. So here we have taken one example. A is a matrix of order 2 cross 3 and B is a matrix of order 3 cross 2. Then the matrix multiplication is done, that is row to column. So first row, first column. And next you should take first row, second column. I said row to column. Whenever you see the first matrix see row wise, second matrix you see column wise. So first row first column means A11 into B11 plus corresponding elements students you have to multiply and then you should add it.

[00:11:00]That is what you should do. A11 into B11 plus you should do A12 into B21 plus A13 into B31. This is first row first column. Similarly, first row second column. A11 into B12 plus A12 into B22 A13 into B32. Next comes second row first column. A21 into B11 plus A22 into B21 A23 into B31. Next last second row second column. A21 into B12 plus A22 into B22 plus A23 into B32. And this is a matrix of order what we are getting, that is 2 cross 2. Yes, here we have.

[00:11:45]Similarly, comes that is properties. Matrix multiplication is not actually commutative. In general students, A into B is not equal to B into A. It holds good. It holds good. I mean, commutative law holds good. AB is equal to BA you are getting only for the diagonal matrices of same order. Whenever A and B are diagonal matrices, you know, diagonal matrices means non non diagonal elements must be zero in that. So whenever you have two diagonal matrices of same order, in that case students, A into B is equal to B into A. But in general remember, matrix multiplication doesn't satisfy that is commutative law.

[00:12:24]Next comes associative law holds good here. A into BC is same as AB into C.

[00:12:30]Distributive law also holds good, that is A into bracket B plus C. This can be written as A into B plus A into C. Or if you have A plus B into C, here we say students, matrix multiplication is distributed over addition. Okay? So here AC plus what you can write, that is BC.

[00:12:49]>> [snorts] >> So this is regarding distributive law.

[00:12:52]Next comes multiplicative identity.

[00:12:54]Okay, this is again the very important [clears throat] one.

[00:12:57]We must know that is the multiplicative identity is actually identity matrix.

[00:13:01]Whenever we do students A into I or I into A, the answer is again that is A we will get. So that's why this I is called as multiplicative identity. Remember this line. A into I whenever we do, the answer is again A. So I is called as a multiplicative identity. Okay? This is multiplicative identity. Next comes multiplicative inverse. Okay.

[00:13:28]Multiplicative inverse is nothing but suppose I have a matrix A now, so that I'll multiply students another matrix B such that A into B you do or B into A you do, if the answer is I, then whatever this B matrix is there now, it is called as students inverse of the matrix A. And that is what we write it as a A inverse is equal to B.

[00:13:51]So how to find A inverse? We see later.

[00:13:54]So just you remember now, whenever we do A into B and B into A is equal to I, this matrix B is called as multiplicative inverse. And what is multiplicative identity? That is identity matrix I I.

[00:14:06]A into I is always A and whenever here now, from this what we got is A inverse is nothing but B. B is called as inverse of the matrix. So that's why here what we got is A into A inverse whenever we do, the answer is always that is I. So two things you remember. A into I is always A and whenever we do any matrix into its inverse now, the result is always identity matrix. Next comes that is transpose of a matrix. Whenever you have matrix A as a matrix with the order m cross n, then its transpose is students obtained by interchanging rows and columns. I mean, rows you write it as a columns, that is what you will get transpose of a matrix. That is denoted by always either A dash or A B.

[00:14:55]And [snorts] whenever you do now, that is transpose, the number of rows uh sorry, number rows and columns you are interchanging means so obviously the order will also interchange here. A is a matrix of order m cross n means you will get A dash is a matrix of order that is n cross m.

[00:15:13]And using this students transpose, we are defining two more matrices. One is symmetric and another one is skew symmetric. So whenever A dash is same as A, that matrix is said to be symmetric.

[00:15:24]And symmetric in a sense what happens here, that is whenever we take students any >> [snorts] >> square matrix we take first and the symmetric element means whenever you fold this square paper, this element will coincide with this. Okay? So whatever present here now, for example, if this is A12 and this is A21, then A12 must be same as A21. Then this matrix is called as symmetric. That is what they written here. IJth element must be same as JIth element. 1 2 2 1. So then what we can say is the matrix is symmetric.

[00:16:02]And if it is skew symmetric now, then always you will get students A dash is equal to that is minus A. I mean here, A12 and A21. Whenever they are same, that is what symmetric. And whenever students they are same but with the opposite sign, that is what it becomes skew symmetric. And here one more thing what happens is in skew symmetric now, diagonal elements are always that is zero. Diagonal elements are always that is zero. Remember this.

[00:16:30]Next comes invertible matrix. So whenever students inverse exist for any matrix, then in that case that matrix is said to be. See this A, whatever matrix A is there, this is said to be invertible only when if its inverse exist. Okay? So that is what invertible matrix means.

[00:16:48]>> [snorts] >> Next comes you have a definition regarding that is determinant also.

[00:16:54]So determinant is actually a function which is associated with that is each square matrix with a unique number. For example, if you have yes you know, whenever you solve students any square matrix of order one, suppose I'm having three here and this is your matrix A, then determinant of A is again that is three. Suppose [snorts] I have a matrix of order 2 cross 2, suppose 1 2 3 4 is there, then determinant of A will be four. Product of diagonal elements minus product of non non diagonal elements and that is minus two. Same same thing we are doing even for the 3 cross 3 matrix by expanding along first row. Even this value also you are getting a real number only. So what it shows here is first square matrix sorry, determinant is defined by taking a square matrix and every time students you are getting here a unique real value. And that's why it is called as determinant is actually a function and it is defined. What is first set here?

[00:17:54]Whatever the first set it is there, it contains that is square matrices. And here students, the range is always that is set of [snorts] real numbers. I mean, every time you are getting the real number as a value here. So that's why they say determinant is a real valued function.

[00:18:14]The domain is always the square matrices and the codomain is always that is set of real numbers. Okay? Every square matrix is related to a unique number here. That is what we call a determinant.

[00:18:29]How to find determinant for 1 cross 1, 2 cross 2, 3 cross 3? Yes, we know very well. Next comes that is in determinant students we are defining that is minor of an element, cofactor of an element and as well as adjoint matrix. Minor means whatever the element is there A I J now, that is a determinant [snorts] obtained by deleting that row and that column. If A I J is the element, then leave I throw and Jth column. For the remaining elements students you should find the determinant. How much you will get the determinant value, that is minor. Once you got minor, you substitute in this, you will get students the corresponding cofactor here.

[00:19:11]Minor of an element, cofactor of an element. Cofactor of an element is capital letter A I J and it is defined as minus one raised to I plus J into M I J. M I J is a minor. So once you got minor here, you put, you will get that is cofactor. And all the cofactor students you write in a matrix. Okay?

[00:19:30]Capital letter [snorts] A11, capital letter A12, capital letter A13. Like this you find and you do the transpose of this and that is what we call it as a adjoint of the matrix A. So these are the terms you should remember that is from this matrix and determinants. Only I told you students the terms here. And other than this we have so many standard other than this, we have so many standard results, shortcuts, those know we see in the next video. Thank you.

[00:20:01]Have a great time.