What do the source symbols A_ij, |A_ij| and the cofactor C_ij denote?

Q: What do the source symbols A_ij, |A_ij| and the cofactor C_ij denote?

In the source notation, $A_{ij}$ represents the $2\times 2$ submatrix obtained by deleting the $i$-th row and $j$-th column from the original matrix $A$. The symbol $|A_{ij}|$ denotes the determinant of this submatrix, which is called the minor $M_{ij}$. The cofactor $C_{ij}$ is the signed minor, calculated as $C_{ij}=(-1)^{i+j}M_{ij}$. The bars around $A_{ij}$ indicate a determinant, not an absolute value, so $|A_{ij}|$ can be negative.

In the source notation,

A_{ij}

represents the

2\times 2

submatrix obtained by deleting the

i

-th row and

j

-th column from the original matrix

A

. The symbol

|A_{ij}|

denotes the determinant of this submatrix, which is called the minor

M_{ij}

. The cofactor

C_{ij}

is the signed minor, calculated as

C_{ij}=(-1)^{i+j}M_{ij}

. The bars around

A_{ij}

indicate a determinant, not an absolute value, so

|A_{ij}|

can be negative.

Conditions

A is a 3x3 real square matrix.
i and j belong to {1, 2, 3}.

Reasoning, step by step

Identify $A_{ij}$ as the submatrix formed by removing row i and column j.
Calculate the determinant of this submatrix to find the minor $M_{ij}$ , denoted as $|A_{ij}|$ .
Apply the position sign $(-1)^{i+j}$ to the minor to obtain the cofactor $C_{ij}$ .
Recognize that $|A_{ij}|$ is a scalar value that can be negative, unlike an absolute value.

Example

For the given matrix, deleting row 2 and column 1 leaves a 2x2 submatrix. Its determinant is $|A_{21}|=-8$ . This is the minor $M_{21}$ . The cofactor is $C_{21}=(-1)^{2+1}(-8)=8$ .

Common misconceptions

Confusing the submatrix $A_{ij}$ with its determinant $|A_{ij}|$ .
Interpreting the vertical bars in $|A_{ij}|$ as absolute value signs, leading to the incorrect assumption that minors are always non-negative.
Treating the minor $M_{ij}$ as the cofactor $C_{ij}$ without applying the position sign.

Watch the explanation

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Connected concepts

Determinants →Matrices →

What do the source symbols A_ij, |A_ij| and the cofactor C_ij denote?

Conditions

Reasoning, step by step

Example

Common misconceptions

Watch the explanation

Connected concepts

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Related questions

What do the source symbols A_ij, |A_ij| and the cofactor C_ij denote?

Conditions

Reasoning, step by step

Example

Common misconceptions

Watch the explanation

Connected concepts

Explore next

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