https://file.aiquickdraw.com/mathbot/file/1726772579191yysfnak2.jpg

For the given determinant △, find the minors and the cofactors of the elements  a_{i2}  and  a_{j3} .

Calculate the determinant △:
a) by expanding it along the elements of the i-th row;
b) by expanding it along the elements of the j-th column;
c) by first obtaining zeros in the i-th row.

Let's analyze the image and solve the given tasks step by step.

### Step 1: Identify the matrix and the given indices \( i \) and \( j \).

From the image, the matrix \( A \) is:
\[
A = \begin{pmatrix}
0 & -2 & 1 & 7 \\
4 & -8 & 2 & -3 \\
10 & 1 & -5 & 4 \\
-8 & 3 & 2 & -1
\end{pmatrix}
\]
and the given indices are \( i = 4 \) and \( j = 2 \).

We will perform the following tasks:
1. **Find the minors and cofactors** of the elements \( a_{i2} \) and \( a_{j3} \).
2. **Calculate the determinant**:
   - (a) by expanding it along the elements of the \( i \)-th row,
   - (b) by expanding it along the elements of the \( j \)-th column,
   - (c) by first obtaining zeros in the \( i \)-th row.

### Step 2: Minors and cofactors of \( a_{i2} \) and \( a_{j3} \)

- \( i = 4 \), so \( a_{i2} = a_{42} = 3 \)
- \( j = 2 \), so \( a_{j3} = a_{23} = 2 \)

#### Minor and cofactor of \( a_{i2} = 3 \)
To find the **minor** of \( a_{42} \), we remove the 4th row and 2nd column from matrix \( A \), which gives the following 3x3 submatrix:
\[
\begin{vmatrix

For the given determinant △, find the minors and the cofactors of the elements a_{i2} and a_{j3}. Calculate the determinant △: a) by expanding it along the elements of the i-th row; b) by expanding it along the elements of the j-th column; c) by first obtaining zeros in the i-th row.

This problem involves calculating the minors and cofactors of elements in a 4x4 matrix, expanding the determinant along specific rows and columns, and using matrix properties to simplify the solution. Suitable for Grades 11-12 or early college level.

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