Question

Use the properties of determinants to find a. Do not evaluate the determinants. Answer: a = det -20 -92 16 172 124 68 12 4 -24, = a. det -5 43 3 -23 4 31 17 1 -6

Answer 1

The student's question appears to mix the concepts of determinants and the quadratic formula. The equation starts with determinants but shifts to quadratic equations, though the information given is not sufficient to solve for 'a' using either method.

Step-by-step explanation:

The student's question pertains to the determinants and their properties. The equation provided appears to be a determinant matrix equation intended to solve for the variable a.

However, the components provided after this seem to relate to solving quadratic equations using the quadratic formula, not determinants. This involves another mathematical concept where a, b, and c are coefficients of the quadratic equation at² + bt + c = 0. The solutions for t would be found using the formula: -b ± √(b² - 4ac) / (2a).

Unfortunately, the information given is somewhat disjointed and there seems to be some confusion between the concept of determinants and the quadratic formula. If the original determinant equation were complete and correctly presented, the properties of determinants could be used to find a without evaluating the determinant.

Answer 2

Thus,
`\( a \)` is -64. This means that the determinant of the first matrix is -64 times the determinant of the second matrix without actually calculating the determinants themselves.

1. Observation of Matrix Elements:

When we compare corresponding elements of the two matrices, we observe that each element of the first matrix is -4 times the corresponding element of the second matrix. For example:

`\[ \text{Element from first matrix} = -20 \]`

`\[ \text{Corresponding element from second matrix} = -5 \]`

`\[ -20 = -4 * (-5) \]`

And this relationship holds for every corresponding pair of elements in the two matrices.

2. Determinant Property for Scalar Multiplication:

A known property of determinants is that if you multiply all elements of a single row or column by a scalar, the determinant of the matrix is multiplied by that scalar. If we multiply every element of a matrix by a scalar, we effectively multiply each row by that scalar, which multiplies the determinant by the scalar raised to the power of the number of rows.

3. Application to the Given Matrices:

Since every element of the first matrix is -4 times the corresponding element of the second matrix, we multiply every row of the second matrix by -4 to obtain the first matrix.

4. Effect on the Determinant:

The effect of multiplying every element of a 3x3 matrix by -4 is equivalent to multiplying the determinant of the matrix by \( (-4)^3 \), because there are three rows.

5. Calculation of \( a \):

We calculate \( a \) by raising -4 to the power of 3:

`\[ a = (-4)^3 \]`

`\[ a = -4 * -4 * -4 \]`

`\[ a = 16 * -4 \]`

`\[ a = -64 \]`

Thus,
`\( a \)` is -64. This means that the determinant of the first matrix is -64 times the determinant of the second matrix without actually calculating the determinants themselves.