 Hello and welcome to the session. In this session we will discuss addition and subtraction of matrices. First let's discuss about the addition of matrices. Two matrices can only be added if they are of same order. If suppose we have two matrices A and B which are of same order then some of these two matrices is obtained by adding the corresponding elements of the matrices A and B. So, A plus B is also a matrix of same order as of A and B. Consider a matrix A of order 2 by 2 with elements 1, 2, 3, 4 and a matrix B of order 2 by 2 with elements 2, 1, 4, 3 as these two matrices are of the same order that is they both are of 2 by 2 order. So, there is some that is A plus B would also be a matrix of order 2 by 2 and its elements are obtained by adding the corresponding elements of the two matrices. So, we have A plus B is equal to matrix with elements 3, 3 in the first row, 7, 7 in the second row. Now, the negative of a matrix A is denoted by minus A. The negative of a matrix which is formed by replacing each entry in the matrix with the additive inverse. Like if you consider a matrix A with elements 1, 3, 5, 6 then the negative of this element A is denoted by minus A and it is a matrix which would be formed by replacing each entry of the matrix A with the additive inverse. So, additive inverse of 1 is minus 1, additive inverse of 3 is minus 3, additive inverse of 5 is minus 5 and additive inverse of 6 is minus 6. So, this is the negative of the matrix A which is minus A. Now, sum of the matrix and its negative is a 0 matrix and this is equal to a matrix with all the elements as 0. So, A plus of minus A is equal to a 0 matrix. Next, we discuss subtraction of matrices. Now, two matrices can be subtracted if they are of same order. Consider a matrix A and a matrix B which are of same order then A can be subtracted and A minus B would be equal to A plus of minus B. That is, we add the negative of the matrix B with the matrix A to obtain the matrix A minus B and this A minus B would also be of the same order as the matrices A and B. And also the difference between the two matrices is obtained by subtracting the corresponding entries to the two matrices. Suppose we have a matrix A with elements P, Q, R, S and we have a matrix B with elements A, B, C, D then A minus B is equal to A plus of minus B. Now, let's find out minus B. This is a matrix with elements minus A minus B minus C minus D. Now, A minus B is equal to the matrix with elements P, Q, R, S that is a matrix A plus the matrix minus B which has elements minus A minus B minus C minus D. So, now A minus B is a matrix with elements P minus A, Q minus B, R minus C, S minus D. Now, we shall discuss the properties of addition of matrices. First property is that addition of matrices is commutative that is if A and B are two matrices of same order then A plus B is equal to B plus A. The next property is that addition of matrices is associated. So, it is the associativity property according to which we have that is A, B and C are three matrices of same order then according to this property we have A plus B the whole plus C is equal to A plus B plus C the whole. Next is the existence of additive identity consider this matrix O which is a null matrix and we have a matrix A and the matrix O and matrix A are of same order then matrix A plus the null matrix O is equal to null matrix O plus the matrix A and each is equal to the matrix A and here this null matrix O is the additive identity. Next we have existence of additive inverse if we have a matrix A then matrix A plus its negative that is A plus or minus A is equal to O which is a null matrix and this is same as minus A plus matrix A. So, this matrix minus A is the additive inverse of matrix A. Let us consider one example to verify the above properties of addition of matrices for this we will consider three matrices A B and C of same order let a matrix A be a matrix with elements 7, 2 in the first row, 4, 5 in the second row, matrix B with elements 2, 4 in the first row, 3, 6 in the second row, a matrix C with elements minus 1, 2 in the first row minus 4, 3 in the second row. Now first property is commutative property according to which we have that A plus B is equal to B plus A. Now let us try to prove this property for the given matrices A and B. Now A plus B is a matrix obtained by adding the corresponding elements of the matrices A and B so here 7 plus 2 is 9 so we write here 9, 2 plus 4 is 6, 4 plus 3 is 7, 5 plus 6 is 11 so this is A plus B. Now B plus A would be a matrix obtained by adding the corresponding elements of matrices B and A. Now 2 plus 7 is 9, 4 plus 2 is 6, 3 plus 4 is 7, 6 plus 5 is 11. Now as you can see that A plus B is same as B plus A so we have proved the commutative property. Next is the associative property in which we have A plus B the whole plus C is equal to A plus B plus C the whole. Now you already have A plus B so now A plus B the whole plus C would be equal to matrix with elements 9, 6, 7, 11. This is A plus B plus the matrix C which is matrix with elements minus 1, 2 minus 4. 3. Now on adding these two matrices we get a matrix with elements 8, 8, 3, 14. So this is A plus B the whole plus C. Now let's find out B plus C. Now this is equal to the matrix with elements 2, 4, 3, 6, matrix plus the matrix C with elements minus 1, 2, minus 4, 3. So this is equal to a matrix with elements 1, 6, minus 1, 9. This is B plus C. Now we add matrix A to the matrix B plus C. So this is equal to the matrix with elements 7, 2, 4, 5 plus the matrix B plus C with elements 1, 6, minus 1, 9. This is equal to a matrix with elements 8, 8, 3, 14. This is A plus B plus C the whole. Now A plus B the whole plus C is same as A plus B plus C the whole. So we have proved the associated property. Next is the existence of additive identity which is a null matrix such that the matrix when added with the null matrix is same as null matrix plus the matrix and which is same as the given matrix. So matrix A plus the null matrix is equal to matrix with elements 7, 2, 4, 5 plus the null matrix with elements 0, 0, 0, 0. This is equal to the matrix with elements 7, 2, 4, 5 which is same as the matrix A and by the commutative law we have A plus null matrix is equal to null matrix plus A and so each would be equal to the matrix A. So here this matrix O would be additive identity. Next property that we have is existence of additive inverse according to which we have the matrix A plus minus of matrix A is equal to the null matrix O which is same as minus A plus matrix A and this minus A is the additive inverse of matrix A. So next we have matrix A plus of minus A is equal to the matrix with elements 7, 2, 4, 5 plus minus matrix A that is minus 7, minus 2, minus 4, minus 5 are the elements of matrix minus A and so this would be equal to the matrix with elements 0, 0, 0, 0, 0 which is a null matrix O. So we have A plus of minus A is equal to the null matrix O which is same as minus A plus matrix A. This is by the commutative law and this minus A is the additive inverse. Next we have solving matrix equations. Suppose we are given a matrix equation X plus A is equal to B. We have to find this unknown matrix X. So for this we add the matrix minus A to both sides. So this means we have X plus the null matrix O since we know that A plus of minus A is null matrix O and this is equal to B minus A. Since the matrix A plus its negative is equal to the null matrix, X plus the null matrix O is same as the matrix X and so this is equal to B minus A. So we have found the unknown matrix X equal to B minus A. This is the required solution. Let us consider an example of this in which we are given the matrix equation as X plus matrix of 2 by 2 order with elements 3, 1, 5, 7. This is equal to a 2 by 2 matrix with elements 2, 0, 4, 5. As we know we will now add the negative of this matrix on both sides. So X plus the matrix with elements 3, 1, 5, 7 plus the negative of this matrix that is a matrix with elements minus 3, minus 1, minus 5, minus 7 is equal to a matrix with elements 2, 0, 4, 5 plus the matrix with elements minus 3, minus 1, minus 5, minus 7. So further we have X plus the null matrix with elements 0, 0, 0, 0 matters. On adding these two matrices we get a null matrix and this is equal to a matrix obtained by adding these two matrices which is a matrix with elements minus 1, minus 1, minus 1, minus 2. So this gives us the matrix X with elements minus 1, minus 1, minus 1, minus 2. This is how we solve a matrix equation. So this completes the session. Hope you have understood the addition and subtraction of matrices and solving a matrix equation.