Example 16 If A = [2 3; 0 -2] and B = [-8; 8], find matrix X...

Example 16 If A = [2 3; 0 -2] and B = [-8; 8], find matrix X such that 2AX = B. Solution. Since 2AX = B and B is a 2 x 1 matrix => 2AX is a 2 x 1 matrix => AX is a 2 x 1 matrix, but A is a 2 x 2 matrix => X is a 2 x 1 matrix. Let X = [x; y], then 2AX = B => 2[2 3; 0 -2][x; y] = [-8; 8] => [4 6; 0 -4][x; y] = [-8; 8] => [4x + 6y; 0.x + (-4)y] = [-8; 8] => 4x + 6y = -8 ...(i) and -4y = 8 => y = -2. Putting y = -2 in (i), we get 4x + 6(-2) = -8 => 4x = 12 - 8 = 4 => x = 1. Therefore X = [1; -2]. Example 17 Given matrix A = [4 sin 30° cos 0°; cos 0° 4 sin 30°] and B = [4; 5]. If AX = B, (i) write the order of matrix X. (ii) find the matrix X.

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00:02 Hi, as you see, example number 16 here, we're given a as 2 3 0 minus 2, as 2 3 0 minus 2.

00:16 So that is given as a here and b is given as minus h8.

00:24 So we're given here 2ax is equal to b.

00:31 2ax is equal to b when the final x here, right? so you can write it as 2 times a it will be 2 ,000.

00:39 3 0 minus 2 and then as a matrix x is equal to this is minus 8 8 that is p now the matrix b is given as the order is 2 cross 1 the order for this is 2 cross 2 so what should be the order of x here the order of x should be 2 cross 1 it should be 2 cross 1 so that if we just multiply this matrix with this matrix we get the order as 2 cross 1 and that should be match with the right inside matrix right on the next one more thing you can just bring this two on the right inside it will become here one over two right now this is given as x y if applied now matrix multiplication so this row multiply with this column here so we get it as 2x plus 3 y on this row multiply the same column here we get 0 x minus 2y is equal to this will be minus 8 over 2 that is minus 4 will be 8 over 2 that will be 8 over 2 that that is giving us 4.

01:50 If you compare now the two matrices here, so we get 2x plus 3y equals minus 4, and we get minus 2y is equal to 4.

02:05 So we get y is equal to minus 2.

02:09 You get it on y -go to minus 2, we're going to put that here, so we'll have 2x minus 6.

02:15 We're going to put that here, minus 6 is equal to minus 4.

02:19 So 2x equals 2 or x is equal to 1.

02:22 So you've already x equal to 1, y is minus 2, the matrix x are coming out to be 1 minus 2.

02:29 That is the required matrix that is given as 1 minus 2 here.

02:39 Next we do now on the example 17.

02:47 So in example 17, we're given the matrix a as 4 sine 30.

02:51 Matrix a is given as 4, sine 30 and then cos 0, then cost 0 and 4 sine 30.

03:10 So first we solve this we get a as will be, and that is 1 over 2.

03:15 So 4 over 2, that is giving as 2...

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