Minors of Matrix Entries

What a Minor Records

Each entry of a square matrix has a minor associated with it. To find that minor, remove the entire row containing the entry and the entire column containing it. The entries left over form a smaller square matrix; the minor is the determinant of that smaller matrix. This deletion rule is simple, but the row and column indices must both be tracked carefully.

For an \(n\times n\) matrix with \(n\ge2\), the minor \(M_{ij}\) is associated with the entry in row \(i\), column \(j\). It is a number, not a matrix entry copied from the original matrix. The matrix that remains after deletion is the matrix whose determinant gives that number. The remaining rows and columns stay in their original order.

The indices in \(M_{ij}\) tell us what to delete: the first index identifies the row and the second identifies the column. For example, \(M_{23}\) means delete row 2 and column 3. It does not mean take the entry in row 2, column 3, nor does it mean delete only that entry.

Minors in a \(3\times3\) Matrix

Deleting one row and one column from a \(3\times3\) matrix leaves a \(2\times2\) matrix. Its determinant can be evaluated using the formula from the earlier material on determinants of \(2\times2\) matrices: \(\det\begin{bmatrix}a&b\\c&d\end{bmatrix}=ad-bc\). Thus, every \(3\times3\) minor requires two steps: identify the four remaining entries in their original order, then apply this formula.

Proof. Deleting row \(i\) leaves exactly two rows, which we call \(r\) and \(s\) in their original order. Deleting column \(j\) leaves exactly two columns, called \(p\) and \(q\) in their original order. The remaining matrix is therefore \(\begin{bmatrix}a_{rp}&a_{rq}\\a_{sp}&a_{sq}\end{bmatrix}\). Applying the \(2\times2\) determinant formula gives \(a_{rp}a_{sq}-a_{rq}a_{sp}\), as claimed. \(\square\)

For a \(3\times3\) matrix, the general rule can also be written out entry by entry. Each expression uses the four entries left after the specified row and column are removed:

For instance, \(M_{23}\) deletes row 2 and column 3, leaving rows 1 and 3 and columns 1 and 2. Its remaining matrix is \(\begin{bmatrix}a_{11}&a_{12}\\a_{31}&a_{32}\end{bmatrix}\), so \(M_{23}=a_{11}a_{32}-a_{12}a_{31}\). This entry-by-entry form is a useful check on indexing, but the deletion procedure is the reliable way to construct each minor.

Minors in a \(4\times4\) Matrix

Deleting one row and one column from a \(4\times4\) matrix leaves a \(3\times3\) matrix. The minor is the determinant of that matrix, not the matrix itself. To calculate numerical \(4\times4\) minors, we need to evaluate the resulting \(3\times3\) determinants. For a matrix \(\begin{bmatrix}a&b&c\\d&e&f\\g&h&k\end{bmatrix}\), the six-term formula is \(aek+bfg+cdh-ceg-bdk-afh\).

Proof. In the permutation definition of a determinant, each term chooses one entry from each row and each column. For three rows, the six possible column orders are \((1,2,3),(2,3,1),(3,1,2),(3,2,1),(2,1,3),(1,3,2)\). The first three orders have positive sign and give the products \(aek\), \(bfg\), and \(cdh\). The last three have negative sign and give \(ceg\), \(bdk\), and \(afh\). Adding the positive products and subtracting the negative products gives the displayed formula. \(\square\)

A Reliable Deletion Procedure

When a minor is requested, it helps to separate the selection of the submatrix from its determinant calculation. This avoids mixing up the original entry \(a_{ij}\) with the number \(M_{ij}\). Use the following sequence:

This procedure also gives a useful size check. A minor from a \(3\times3\) matrix must come from a \(2\times2\) matrix, so it is calculated as a \(2\times2\) determinant. A minor from a \(4\times4\) matrix must come from a \(3\times3\) matrix. If the submatrix has the wrong size, at least one deletion was not carried out correctly.

Common Mistake: Deleting Only the Entry

Consider \(\mathbf{C}=\begin{bmatrix}2&1&4\\3&5&0\\1&2&6\end{bmatrix}\), and suppose we need \(M_{12}\). A wrong approach is to remove only the entry \(c_{12}=1\) and then choose nearby entries to form a \(2\times2\) determinant. For example, \(\det\begin{bmatrix}2&4\\3&0\end{bmatrix}=-12\) is not \(M_{12}\): that matrix has not been formed by deleting row 1 and column 2.

The correct deletion removes all of row 1 and all of column 2. The remaining entries are from rows 2 and 3 and columns 1 and 3:

The wrong calculation produced \(-12\), while the correct minor is \(18\). The submatrix itself is the check: its entries must come only from the rows and columns that survive both deletions.

Why Minors Are Useful

Minors turn a determinant problem into a smaller determinant problem. For the examples here, the definition reduces \(3\times3\) matrices to \(2\times2\) determinants and \(4\times4\) matrices to \(3\times3\) determinants. This reduction is the reason minors are introduced before determinant calculations involving larger matrices. At this stage, the essential skill is to form the right smaller matrix and evaluate its determinant accurately.

Keep three distinctions in mind. The entry \(a_{ij}\) is a scalar in the original matrix. The submatrix used for \(M_{ij}\) is formed after deleting row \(i\) and column \(j\). The minor \(M_{ij}\) is the determinant of that submatrix. Deletion itself does not insert an extra sign or alter any surviving entry; it only selects the smaller matrix in its original order.