The Linearity Of Determinants: Exploring The Relationship Of 2X2 Matrices

the det of a 2×2 matrix is a linear function of each row of the matrix when the other row is held fixed

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Let’s consider a 2×2 matrix A = [[a, b], [c, d]]. The determinant of this matrix is given by:

det(A) = ad – bc

To understand the statement that the determinant is a linear function of each row of the matrix when the other row is held fixed, we can consider two cases:

Case 1: We fix the first row and vary the second row

In this case, we hold the first row fixed as [a, b] and vary the second row as [x, y]. The matrix becomes:

A = [[a, b], [x, y]]

The determinant of this matrix is:

det(A) = ay – bx

Now, we can see that the determinant is a linear function of the second row [x, y] with coefficients a and -b. The first row [a, b] is held fixed.

Case 2: We fix the second row and vary the first row

In this case, we hold the second row fixed as [c, d] and vary the first row as [u, v]. The matrix becomes:

A = [[u, v], [c, d]]

The determinant of this matrix is:

det(A) = ud – cv

Now, we can see that the determinant is a linear function of the first row [u, v] with coefficients d and -c. The second row [c, d] is held fixed.

Therefore, we can conclude that the determinant of a 2×2 matrix is a linear function of each row of the matrix when the other row is held fixed.

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