 Hi and welcome to the session. I am Shashi and I am going to help you to solve the following question. Question is find the matrix X so that X multiplied by matrix 1, 2, 3, 4, 5, 6 is equal to matrix minus 7, minus 8, minus 9, 4, 5, 6. First of all let us understand that if A is a matrix of order M into M and B is a matrix of order N into B, then the product AB is defined and the order of AB is M into P. Since the number of columns of A is equal to number of rows of B, so the product of A and B is defined and the order of AB is M multiplied by P. Now let us start with the solution. We are given matrix X multiplied by matrix 1, 2, 3, 4, 5, 6 is equal to matrix minus 7, minus 8, minus 9, 2, 4, 6. Now from key idea we know the matrix X must have rows equal to a number of rows of this matrix that is equal to 2 as this matrix is having 2 rows. Similarly the number of columns of X must be equal to number of rows of this matrix. Now number of rows of this matrix is 2, so the number of columns of X must be equal to 2. So X is a matrix of order 2 into 2. Now let matrix X is equal to X1, X2, X3, X4. Now we will replace this X by this matrix. So we get X1, X2, X3, X4 is equal to matrix 1, 2, 3, 4, 5, 6 is equal to matrix minus 7, minus 8, minus 9, 2, 4, 6. Now multiplying the 2 matrices on the left hand side we get 1 multiplied by X1 plus 4 multiplied by X2, 2 multiplied by X1 plus 5 multiplied by X2, 3 multiplied by X1 plus 6 multiplied by X2, 1 multiplied by X3 plus 4 multiplied by X4, 2 multiplied by X3 plus 5 multiplied by X4, 3 multiplied by X3 plus 6 multiplied by X4 is equal to matrix minus 7, minus 8, minus 9, 2, 4, 6. Now further simplifying we get matrix x1 plus 4x2, 2x1 plus 5x2, 3x1 plus 6x2, x3 plus 4x4, 2x3 plus 5x4, 3x3 plus 6x4 is equal to matrix minus 7, minus 8, minus 9, 2, 4, 6. Now comparing the corresponding elements we get x1 plus 4x2 is equal to minus 7, then 2x1 plus 5x2 is equal to minus 8, 3x1 plus 6x2 is equal to minus 9, x3 plus 4x4 is equal to 2, 2x3 plus 5x4 is equal to 4, 3x3 plus 6x4 is equal to 6. Let us name these equations as equation 1, equation 2, equation 3, equation 4, equation 5 and equation 6. Now multiplying equation 1 by 2 and subtracting from equation 2 we get 2x1 plus 5x2 is equal to minus 8, minus 2x1 minus 8x2 plus 14. We have reduced this equation by multiplying equation 1 by 2. Now subtracting the two equations we get 2x1 and 2x1 we get cancelled, we get minus 3x2 is equal to plus 6. This implies x2 is equal to 6 upon minus 3 which is equal to minus 2. So therefore x2 is equal to minus 2. Now we will substitute this value of x2 in equation 1 to get the value of x1. So substituting the value of x2 is equal to minus 2 in equation 1 we get x1 plus 4 multiplied by minus 2 is equal to minus 7. This implies x1 minus 8 is equal to minus 7. This implies x1 is equal to minus 7 plus 8. This further implies x1 is equal to 1. Therefore we get x1 is equal to 1. Now multiplying equation 4 by 2 and subtracting it from equation 5 we get. Now multiplying equation 4 by 2 we get 2x3 plus 8x4 is equal to 4. And the equation 5 is 2x3 plus 5x4 is equal to 4. Now we will subtract the two equations. So we will change the signs to subtract the two equations and get 2x3 and 2x3 get cancelled. We get minus 3x4 is equal to 0. This implies x4 is equal to 0. Therefore x4 is equal to 0. Now we will substitute this value of x4 is equal to 0 in the equation 4 to get the value of x3. So we get x3 plus 4 multiplied by 0 is equal to 2. This implies x3 is equal to 2. Therefore x3 is equal to 2. Hence we get x1 is equal to 1. x2 is equal to minus 2. x3 is equal to x4 is equal to 0. Therefore matrix x is equal to matrix 1 minus 2 to 0. We had substituted the values of x1, x2, x3 and x4 in the matrix x. So this is our required answer. This completes the session. Hope you understood the session. Take care and goodbye.