What is the value of a that would make the matrix equation true?
(1 -2) (-5 6) = (a b)
(3 4) (7 -8) = (c d)
To find the value of "a" that would make the matrix equation true, we need to multiply the matrices on the left-hand side:
⎛1 -2⎞ ⎛-5 6⎞ = ⎛a b⎞
⎜3 4⎟ ⎜7 -8⎟ ⎜c d⎟
To multiply matrices, we need to follow a set of rules.
1. The number of columns in the first matrix should be equal to the number of rows in the second matrix.
In this case, the first matrix has 2 columns and the second matrix has 2 rows, so we can proceed with the multiplication.
2. Multiply each element in the first row of the first matrix by the corresponding element in the first column of the second matrix. Sum these products to get the first element of the resulting matrix.
For the first element in the resulting matrix, we multiply 1 by -5 and -2 by 7, then add these products: (1 * -5) + (-2 * 7) = -5 - 14 = -19.
3. Repeat step 2 for the second element of the resulting matrix. Multiply each element in the first row of the first matrix by the corresponding element in the second column of the second matrix. Sum these products to get the second element of the resulting matrix.
For the second element in the resulting matrix, we multiply 1 by 6 and -2 by -8, then add these products: (1 * 6) + (-2 * -8) = 6 + 16 = 22.
Now we have the resulting matrix:
⎛-19 22⎞ = ⎛a b⎞
⎜? ? ⎟ ⎜c d⎟
To make this matrix equation true, the values of "a," "b," "c," and "d" on the right-hand side should correspond to the elements of the resulting matrix.
In this case, the first element of the resulting matrix is -19, so "a" must be -19. The second element of the resulting matrix is 22, so "b" must be 22.
Therefore, the value of "a" that would make the matrix equation true is -19.