 Hello and welcome to the session. In this session we will discuss the properties of matrix multiplication. First of all, let's discuss the identity element for multiplication of two matrices. So, a unit matrix is generally denoted by I and if we have a unit matrix of order 2 then we can write it as I2. Then this would be a matrix in which the diagonal elements are 1 and rest of the elements are 0. And suppose we have a matrix A with elements A, B, C, D then the matrix A into the unit matrix I2 of order 2 would be equal to a matrix with elements A, B, C, D. That is on multiplying these two matrices we obtain a matrix with elements A, B, C, D and this is same as the matrix A. In the same way when we multiply the unit matrix I2 with the matrix A we would get a matrix with elements A, B, C, D. This is also same as the matrix A. So we have A into I2 is equal to I2 into A is equal to A. This I2 that is the unit matrix I2 is the identity matrix or you can say that it is the identity element for multiplication in the set of square matrices of order 2. Let's start with the properties of matrix multiplication. First we have product of matrices is not commutative that is if we have matrix A and matrix B then their product that is AB is not equal to BA. Let us now consider a matrix A with elements in the first row as 2, 1 in the second row as minus 1, 2 and consider a matrix B with elements in the first row as 1, 5 in the second row the elements are 4, 8. Now AB is equal to the matrix with elements 2, 1, minus 1, 2 multiplied by the matrix B with elements 1, 5, 4, 8. Now on multiplying these two matrices we get a matrix with elements 6, 18, 7, 11. Let's now find out BA. Now BA would be equal to the matrix B with elements 1, 5, 4, 8 into the matrix A with elements 2, 1, minus 1, 2 on multiplying these two matrices we get a matrix with elements minus 3, 11, 0, 20. Now we observe that AB is not equal to BA. This shows that product of matrices is not commutative. Now the next property is multiplication of matrices is associative. This means if we have three matrices AB and C where AB is defined and BC is also defined then AB the whole into C is equal to A into BC the whole. Let's consider the same matrices A and B and consider a matrix C with elements 2, 4, 6, 8. Now we have already found out AB. Now let's find out BC. Now BC is equal to the matrix B with elements 1, 5, 4, 8 into the matrix C with elements 2, 4, 6, 8. Now on multiplying these two matrices we get a matrix. Here we have 32, 44, 56, 80. Now let's find out AB the whole into C. This is equal to the matrix AB which is this. There is a matrix with elements 6, 18, 7, 11 into the matrix C with elements 2, 4, 6, 8. Now on multiplying these two matrices we get a matrix with elements 1, 20, 168, 80, 116. Now we will find out A into BC the whole. This is equal to the matrix A with elements 2, 1, minus 1, 2 into the matrix BC with elements 32, 44, 56, 80. On multiplying these two matrices we get a matrix with elements 1, 20, 168, 80, 116. So we observe that these two matrices are equal. That is we have AB the whole into C is equal to A into BC the whole. This shows that the multiplication of matrices is associative. Let's now discuss the next property which says that the product of two matrices can be a zero matrix without either factor being a zero matrix. That is if we have two matrices A and B then AB would be equal to the zero matrix O even if we have A is not a zero matrix and B is also not a zero matrix. Consider a matrix A which is not a zero matrix with elements 2, minus 2, 4, minus 4 and also consider a matrix B which is also not a zero matrix with elements 2, 5, 2, 5. Now let's find out AB that is the product of the two matrices A and B. This would be equal to a matrix with all the elements as zero matrix. We get AB as a zero matrix and we have that A and B are not zero matrices. Next property. According to this property we have that the cancellation law for the multiplication real numbers is not valid for the multiplication of matrices. If AB is equal to AC then this does not imply that B is equal to C. Consider a matrix A with elements 2, minus 2, 4, minus 4, a matrix B with elements 5, 6, 4, 4 and also with elements 3, 8, 2. Now find out AB this is equal to the matrix with elements 2, minus 2, 4, minus 4 into the matrix with elements 5, 6, 4, 4. That is the matrix B. On multiplying these two matrices we get a matrix with elements 8. Let's now find out AC is equal to the matrix with elements 2, minus 2, 4, minus 4. That is the matrix A into the matrix C with elements 3, 8, 2, 6. On multiplying these two matrices we get a matrix with elements 4, 4, 8. So we observe that AB is same as AC but B is not equal to C. So this does not imply that B is equal to C. That is we cannot cancel A from these two sides. Next property, multiplication matrices is distributive with respect to matrix addition. That is if we have three matrices A, B and C of same order then matrix A into B plus C the whole is equal to AB plus AC and also B plus C the whole into A is equal to BA plus CA. Again consider a matrix A with elements 2, 1, minus 1, 2, a matrix B with elements 1, 5, 4, 8. A matrix C with elements 2, 4, 6, 8. Let's now find out A into B plus C the whole. This is equal to the matrix A with elements 2, 1, minus 1, 2. This into matrix B with elements 1, 5, 4, 8 plus the matrix C with elements 2, 4, 6, 8. That's the whole. So further this is equal to the matrix with elements 2, 1, minus 1, 2. This into the matrix that we obtain by adding these two matrices. So when we add these two matrices we get a matrix with elements 3, 9, 10, 16. Now further when we multiply these two matrices we get a matrix with elements 16, 34, 17, 23. This matrix is A into B plus C the whole. Let's now find out AB. This is equal to the matrix A with elements 2, 1, minus 1, 2 into the matrix B with elements 1, 5, 4, 8. On multiplying these two matrices we get a matrix with elements 6, 18, 7, 11. This is matrix AB. Now in the same way let's find out the matrix AC. This is equal to the matrix A with elements 2, 1, minus 1, 2 into the matrix C with elements 2, 4, 6, 8. When we multiply these two matrices we get a matrix with elements with elements in the first row as 10, 16 in the second row as 10, 12. Now let's add AB and AC. So AB plus AC is equal to a matrix with elements 16, 34, 17, 23. Now we observe that A into B plus C the whole is same as AB plus AC. So A into B plus C the whole is equal to AB plus AC that is the matrix multiplication is distributed with respect to matrix addition. In the same way we can also show that A plus B the whole into the matrix C is equal to AC plus BC. So this completes the session. Hope you understood the properties of matrix multiplication.