DLA Reflect on your understanding

Reflections 8.7 Reflect on your understanding

The reflections activities in this section concern only square matrices.

In addition to the reflection activities below, re-read Section 8.2 Terminology and notation. Be sure you understand each of the new definitions introduced in this chapter, and spend some time committing them to memory.

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1. $1 \times 1$ determinants.

Explain the conceptual reason we have defined the determinant of the general

1 \times 1

matrix

[\begin{matrix} a \end{matrix}]

to be the value of the single entry

a .

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2. Inductive definition of the determinant.

Describe how the definition of the determinant for an

n \times n

matrix depends on the determinant for

(n - 1) \times (n - 1)

matrices (assuming

n \geq 2

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3. Minors versus cofactors.

Describe the relationship between the

{(i, j)}^{th}

minor and the

{(i, j)}^{th}

cofactor of a matrix for a specific pair of indices

(i, j) .

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4. Minors/cofactors versus entries.

(a)

State which entries in an

n \times n

matrix are involved in the computation of the

{(i, j)}^{th}

minor of that matrix. Repeat for the computation of the

{(i, j)}^{th}

cofactor.

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(b)

In particular, do the values of the

{(i, j)}^{th}

minor and cofactor of a matrix depend on the value of the

{(i, j)}^{th}

entry of that matrix?

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5. Cofactor signs.

Describe in words the general pattern of cofactor signs in an

n \times n

matrix.

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6. Determinants of diagonal/triangular matrices.

(a)

State the simple pattern for computing the determinant of a diagonal or upper/lower triangular matrix.

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(b)

What conclusion(s) can be made about a diagonal or triangular matrix whose determinant is known to be zero?

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