Which values in the first analysis interval are minor determinants, and how are cofactor signs applied?

Q: Which values in the first analysis interval are minor determinants, and how are cofactor signs applied?

In the first interval, the values calculated for $|A_{11}|$, $|A_{21}|$, $|A_{31}|$, and $|A_{12}|$ are minor determinants ($M_{ij}$). Specifically, $M_{11}=-14$, $M_{21}=-8$, $M_{31}=1$, and $M_{12}=-17$. Cofactor signs are applied by multiplying each minor by $(-1)^{i+j}$. For example, the cofactor $C_{21}$ is $(-1)^{2+1} \times M_{21} = -1 \times (-8) = 8$. These signed cofactors are then arranged and transposed to form the adjugate matrix.

In the first interval, the values calculated for

|A_{11}|

|A_{21}|

|A_{31}|

, and

|A_{12}|

are minor determinants (

M_{ij}

). Specifically,

M_{11}=-14

M_{21}=-8

M_{31}=1

, and

M_{12}=-17

. Cofactor signs are applied by multiplying each minor by

(-1)^{i+j}

. For example, the cofactor

C_{21}

(-1)^{2+1} \times M_{21} = -1 \times (-8) = 8

. These signed cofactors are then arranged and transposed to form the adjugate matrix.

Conditions

A is the given 3x3 matrix.
The analysis covers the calculation of the first few minors.

Reasoning, step by step

Identify the calculated values $|A_{ij}|$ as minor determinants $M_{ij}$ .
Recall the formula for cofactors: $C_{ij} = (-1)^{i+j} M_{ij}$ .
Apply the sign $(-1)^{i+j}$ to each minor to find the corresponding cofactor.
Note that these cofactors will later be transposed to construct the adjugate matrix.

Example

The screen displays $|A_{21}|=-8$ . This is the minor $M_{21}$ . The cofactor is $C_{21} = (-1)^{2+1}(-8) = 8$ . The later adjugate construction explicitly applies this sign.

Common misconceptions

Assuming that the values $|A_{ij}|$ shown on the screen are already the cofactors.
Forgetting that the minor $M_{21}$ is negative, and incorrectly calculating the cofactor as $-8$ instead of $8$ .
Confusing the minor determinant with the absolute value of the determinant.

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Connected concepts

Determinants →Matrices →

Which values in the first analysis interval are minor determinants, and how are cofactor signs applied?

Conditions

Reasoning, step by step

Example

Common misconceptions

Watch the explanation

Connected concepts

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Related questions

Which values in the first analysis interval are minor determinants, and how are cofactor signs applied?

Conditions

Reasoning, step by step

Example

Common misconceptions

Watch the explanation

Connected concepts

Explore next

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