Consider the following matrix with one empty block in the lower extreme corner:

Correct Option (3)

The matrix follows a consistent pattern for the arrangement of 'O' (circles) and '∆' (triangles) across both rows and columns. To determine the figure for the empty block in the lower extreme corner (R3C3), these patterns must be applied:

• Row-wise Pattern:
  • The number of 'O's decreases from left to right.
  • The number of '∆'s increases from left to right.
• Column-wise Pattern:
  • The number of 'O's decreases from top to bottom.
  • The number of '∆'s decreases from top to bottom.

Therefore, the figure in the empty block (R3C3) must exhibit the following characteristics relative to its adjacent elements:

• The count of 'O's must be less than the 'O's in R3C2 and R2C3.
• The count of '∆'s must be greater than the '∆'s in R3C2 but less than the '∆'s in R2C3.

Option (3) is the only figure that satisfies all these conditions, maintaining the established decreasing trend for 'O's in both its row and column, the increasing trend for '∆'s in its row, and the decreasing trend for '∆'s in its column.

Incorrect Options:

Options (1), (2), and (4) are incorrect because they fail to adhere to one or more of the established patterns within the matrix:

• Some options might show an increasing number of 'O's in a row or column, violating the decreasing trend for 'O's.
• Other options might present a decreasing number of '∆'s in a row, contrary to the required increasing trend.
• Alternatively, certain options could display an increasing number of '∆'s in a column, which would contradict the established decreasing trend for '∆'s in columns.
• Any figure that does not simultaneously satisfy all four conditions (O's decreasing row-wise, ∆'s increasing row-wise, O's decreasing column-wise, ∆'s decreasing column-wise) cannot complete the matrix.