Need help with these--
40. The graph of the following system yields perpendicular lines:
x + 2y = -10
-4y = 2x + 20
True or False?
42. If we multiply a 2 x 2 matrix with a 2 x 1 matrix, the product is a 2 x 1 matrix.
-False?
Thanks
-MC
41. False. If you solve both equations for "y", you will get the same line; which means they can not be perpendicular.
42. True. When you have two matrices multiplied together, (2 x 2)(2 x1). If the two inner numbers match, then your resulting matrix will be the two outer numbers.
Thank you!
-MC
Sure, I'd be happy to help!
40. To determine if the graph of the given system yields perpendicular lines, we need to examine the slopes of the lines. The equations can be rearranged into slope-intercept form (y = mx + b) by solving for y:
For the first equation:
x + 2y = -10
2y = -x - 10
y = -0.5x - 5
For the second equation:
-4y = 2x + 20
4y = -2x - 20
y = -0.5x - 5
To determine the slope of a line given an equation in slope-intercept form, we can look at the coefficient of x (m). In this case, both equations have a slope (m) of -0.5.
If two lines are perpendicular, their slopes are negative reciprocals of each other. The negative reciprocal of -0.5 is 2. Since the slopes of the given equations are the same (-0.5), the lines are not perpendicular.
Therefore, the statement "The graph of the following system yields perpendicular lines" is False.
42. When multiplying matrices, the number of columns in the first matrix must be equal to the number of rows in the second matrix. In this case, we are multiplying a 2 x 2 matrix with a 2 x 1 matrix.
For matrix multiplication, we multiply the corresponding elements of each row in the first matrix with each column in the second matrix and sum up the results to get the resulting matrix. Since the first matrix has 2 columns and the second matrix has 2 rows, we can perform the multiplication.
However, the resulting matrix from multiplying a 2 x 2 matrix with a 2 x 1 matrix is actually a 2 x 1 matrix. Each element of the resulting matrix is obtained by multiplying the corresponding elements of the first matrix with the second matrix and summing them up.
Therefore, the statement "If we multiply a 2 x 2 matrix with a 2 x 1 matrix, the product is a 2 x 1 matrix" is True.
I hope this clarifies your questions! Let me know if you have any more.