If A=(aij) and B=(bij) are arbitrary square matrices of order 2, show that |AB|=|A||B|?
I don't understand how to start this question. What is the question even asking?
The question is asking you to prove that the determinant of the product of two square matrices of order 2 (A and B) is equal to the product of their determinants. In other words, you need to show that |AB| = |A||B|.
To start the proof, you can begin by expanding both sides of the equation using the definition of the determinant. The determinant of a 2x2 matrix A can be computed as follows:
|A| = a11 * a22 - a12 * a21
Similarly, the determinant of a 2x2 matrix B can be computed as:
|B| = b11 * b22 - b12 * b21
Now, let's compute the determinant of the product AB:
|AB| = (a11b11 + a12b21) * (a22b22 + a21b12) - (a11b12 + a12b22) * (a21b11 + a22b21)
Next, we'll simplify the expression:
|AB| = a11b11a22b22 + a11b11a21b12 + a12b21a22b22 + a12b21a21b12 - a11b12a21b11 - a11b12a22b21 - a12b22a21b11 - a12b22a22b21
Now, if we group the terms, we can see that the expression can be written as:
|AB| = (a11a22 - a12a21)(b11b22 - b12b21)
Notice that the expression in parentheses on the right side is actually the product of the determinants of A and B:
|AB| = |A||B|
This shows that the determinant of the product of two square matrices of order 2 is equal to the product of their determinants. Hence, |AB| = |A||B|.
The question is asking you to prove that the determinant of the product of two arbitrary 2x2 matrices, A and B, is equal to the product of their determinants.
To start this proof, you need to recall the formula for the determinant of a 2x2 matrix. Given a matrix A = [[a11, a12], [a21, a22]], the determinant of A, denoted |A| or det(A), is calculated as follows:
|A| = a11 * a22 - a12 * a21
Now, let's consider two arbitrary matrices A and B:
A = [[a11, a12], [a21, a22]]
B = [[b11, b12], [b21, b22]]
The product of A and B, denoted AB, is calculated as follows:
AB = [[a11*b11 + a12*b21, a11*b12 + a12*b22],
[a21*b11 + a22*b21, a21*b12 + a22*b22]]
To prove that |AB| = |A||B|, we need to compute each of these determinants.
1. Compute |A|:
|A| = a11 * a22 - a12 * a21
2. Compute |B|:
|B| = b11 * b22 - b12 * b21
3. Compute |AB|:
|AB| = (a11*b11 + a12*b21) * (a21*b12 + a22*b22) - (a11*b12 + a12*b22) * (a21*b11 + a22*b21)
Your task is to expand this expression and simplify it to show that |AB| = |A||B|. Remember to use the properties of real numbers, such as the distributive property and commutative property, to help you simplify the expression.