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Chapter 3: Q8P (page 95)

Show that if, in using the Laplace development, you accidentally multiply the elements of one row by the cofactors of another row, you get zero. Hint: Consider Fact 2b

Short Answer

Expert verified

It has been proved that zero is obtained.

Step by step solution

01

Results used

The fact used about determinants: The determinant is zero if two rows are identical.

02

Show that the determinant is zero

Suppose, in a Laplace transformation, you accidentally multiply elements of one row by the cofactors of another row.

Now, to find the determinant: We multiply each element of one row by its cofactor and add the results.

Here, if elements of one row are multiplied by cofactors of another, then two rows become identical.

If two rows are identical then the determinant is zero as per the fact given.

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Most popular questions from this chapter

Find the symmetric equations (5.6) or (5.7) and the parametric equations (5.8) of a line, and/or the equation (5.10) of the plane satisfying the following given conditions.

Line through and parallel to the line .

Answer

The symmetric equations of the line is .

The parametric equation is .

Step-by-Step Solution

Step 1: Concept of the symmetric and parametric equations

The symmetric equations of the line passing through and parallel to is

The parametric equations of the line are

Step 2: Determine the symmetric equation of a straight line

The given point is and the line is .

The given line is in the form of . So, we get

The symmetric equations of the straight line passing through and parallel to is given by

Thus, the required solution is .

Step 3: Determine the parametric equation of a straight line.

The parametric equations of the straight line passing through and parallel to is given by

Or

.

Thus, the required solution is .

The Pauli spin matrices in quantum mechanics are $A = (\begin{matrix} 1 & 0 \\ 0 & 1 \end{matrix})$ , $B = (\begin{matrix} 0 & - i \\ i & 0 \end{matrix})$ , $C = (\begin{matrix} 1 & 0 \\ 0 & - 1 \end{matrix})$ .For the Pauli spin matrix C , find the matrices $sinkC$ , $coskC$ , $e^{kC}$ , and $e^{ikC}$ . Hint: Show that if a matrix is diagonal, say $D = (\begin{matrix} a & 0 \\ 0 & b \end{matrix})$ , then $f (D) = (\begin{matrix} f (a) & 0 \\ 0 & f (b) \end{matrix})$ .

Find the symmetric equations $(5.6) or 5.7)$ and the parametric equations $(5.8)$ of a line, and/or the equation $(5.10)$ of the plane satisfying the following given conditions.

Line through $(5, - 4, 2)$ and parallel to the line $r = i - j + (5 i - 2 j + k) t .$

Find the inverse of the transformation $x^{'} = 2 x - 3 y, y^{'} = x + y$ , that is, find x, y in terms of $x^{'}, y^{'}$ .

Show that the following matrices are Hermitian whether Ais Hermitian or not: $A A^{t}, A + A^{t}, i (A - A^{t})$ .

See all solutions

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