 Hello and welcome to this session. In this session we will discuss transformations using matrices. Now in our earlier session we had discussed about translation and variation of figures using matrices. In this session we will engineer it by discussing reflection, rotation of geometric figures on a coordinate plane using matrices. First of all let us discuss reflection. Now reflection is a transformation that creates a symmetry on the coordinate plane and we know in a reflection that pre-image is reflected through the line of reflection. Now as you see how we can reflect a geometric figure in x-axis, y-axis in the line, y is equal to x using matrices. Let us discuss reflection in x-axis. Now to reflect any figure in x-axis on a coordinate plane we multiply the vertex matrix in first row as 1, 0 and elements in second row as 0 minus 1 which is of the reflected image from the matrix obtained. Now let us discuss reflection in y-axis. To reflect any figure in y-axis on a coordinate plane we multiply the vertex matrix and the matrix in first row as minus 1, 0 and elements in second row as 0 and the image from the matrix obtained. Now let us discuss reflection in the line y is equal to x. Now to reflect any figure in the line y is equal to x on a coordinate plane we multiply vertex matrix, y matrix in first row as 0, 1 and elements in second row as 1, 0 and we get the vertices of the reflected image. Now in this table you can see when we have to reflect a geometric figure in x-axis on a coordinate plane then we multiply the vertex matrix on the left by this matrix that is the matrix with elements in first row as 1, 0 and elements in second row as 0 minus 1. To reflect a geometric figure y is equal to x we multiply the vertex matrix on the left by these matrices respectively Now suppose we have a triangle A, B, C with vertices having coordinates 1, 2, B with coordinates 3, 4 and C with coordinates 2 minus 1, y is equal to x. Now first of all let us write its vertex matrix in vertex matrix in first row we will write of all the vertices that will write y coordinates of all these vertices so this is the matrix with elements in first row as 1, 3 and second row as 2 minus 1. Now we have to reflect it in the line y is equal to we will multiply the vertex matrix by so let us multiply the vertex matrix by the matrix with elements in first row as 0, 1 and elements in second row as 1, 0 taking this row that is first row respectively we will get the first element of the first row of the vertex matrix which will be 0 into 1 plus 1 into 2 then we will take first row of vertex matrix and the first row of vertex matrix which will be 0 into 3 plus 1 into 4 Now next we will take first row matrix, third element of the first row of vertex matrix which will be 0 into 2, 1 into minus 1 here we have 1 into 1 plus 0 into 2 and here we have 1 into 3 plus 0 into 4 then again we take matrix and here we have 1 into 2 plus 0 into minus 1 further and simplifying this is equal to matrix with elements in first row as 0 plus 2 0 plus 4, 0 minus 1, 1 plus 0, 3 plus 0 and 2 is equal to matrix with elements in first row as 2, 4 and minus 1 and elements in second row as 1, 3 and now this resultant matrix will give us vertices of the reflective image of triangle A, B, C so we have a reflected image of triangle A, B, C that is triangle A dash, B dash, C dash with vertices with coordinates to then B dash with coordinates with coordinates minus these points on the coordinate plane now when we grab these points on the coordinate plane we get triangle A dash, B dash, C dash which is the reflective image of triangle A, B, C in the line y is equal to x the vertex matrix using, now let us discuss rotation, a rotation as a transformation that turns a figure about a fixed point a free image at an angle of 90 degrees, 180 degrees and 270 degrees about a region in counter clockwise direction using matrices now a geometric figure about a region in counter clockwise direction then we multiply the vertex matrix by the matrix first row as 0 minus 1 and elements in second row as 1, 0 the vertex matrix of the rotated image from it we turn around the vertices with a geometric figure at an angle of 180 degrees about a region in counter clockwise direction then we multiply the vertex matrix elements in first row as minus 1, 0 and elements in second row as 0 minus in the vertex matrix of the rotated image a geometric figure at an angle of origin in counter clockwise direction then we multiply the vertex matrix by the matrix with elements in first row as 0, 1 and elements in second row as minus 1, 0 the vertex matrix of the rotated image now from this table you can see when angle of rotation is 90 degrees, 180 degrees and 270 degrees then we multiply the vertex matrix on the left by these matrices respectively now if we have to rotate this triangle that is triangle ABC's about a region in counter clockwise direction now its vertex matrix will give matrix elements in first row as 1, 3, 2, minus 1 then at 90 degrees about a region in counter clockwise direction we multiply the vertex matrix we will multiply the vertex matrix by the matrix with elements in first row as 0, minus 1 and elements in second row as 1, 0 now on multiplying these two matrices we get a matrix with elements in first row as 0 into 1 of minus 1 into 2 then 0 into 3 plus of minus 1 into 4 and 0 into minus 1 then we have elements in second row as 1 into 1 plus 0 into 2 then 1 into 3 plus 0 into 1 into 2 plus 0 into minus 1 matrix with elements in first row as minus 2, minus 4 and 1 and as 1, 3 with coordinates minus 2, 1 B dash with coordinates minus 4, 3 with coordinates 1, 2 so we have plotted these points on the coordinate plane and when we join these points we get triangle A dash B dash C dash which is the rotated image of triangle ABC so when triangle in counter clockwise B dash C dash can rotate the image variation we have discussed translations using matrices and this completes our session hope you all have enjoyed the session