Finding the Missing Figure in the Matrix

Let's analyze the given matrix to find the pattern and determine the missing figure.

The matrix is:

We need to find the number that replaces the question mark (?). Let's examine the numbers in the rows to see if there is a consistent relationship.

Analyzing Row Patterns

Look at the second row (99, 77, 121). We can observe that these numbers are multiples of 11:

• \(99 = 9 \times 11\)
• \(77 = 7 \times 11\)
• \(121 = 11 \times 11\)

The numbers in the second row follow a pattern: (9 * 11), (7 * 11), (11 * 11).

Now let's look at the third row (135, 105, 165). These numbers appear to be multiples of 15:

• \(135 = 9 \times 15\)
• \(105 = 7 \times 15\)
• \(165 = 11 \times 15\)

The numbers in the third row also follow a similar pattern: (9 * 15), (7 * 15), (11 * 15).

Applying the Pattern to Find the Missing Figure

It seems there is a pattern across the rows where the numbers are (9 * k), (7 * k), and (11 * k), where 'k' is a constant factor specific to that row.

For the first row (63, 49, ?), let's find the factor 'k'.

• We have \(63\) in the first column. Assuming the pattern \(9 \times k\), we get \(63 = 9 \times k\), which means \(k = 63 \div 9 = 7\).
• We have \(49\) in the second column. Assuming the pattern \(7 \times k\), we get \(49 = 7 \times k\), which means \(k = 49 \div 7 = 7\).

The factor 'k' for the first row is consistently 7.

Now, we can find the missing figure in the third column of the first row using the pattern \(11 \times k\):

• Missing figure = \(11 \times 7 = 77\)

Verifying the Missing Figure

Let's place 77 in the matrix and see if the pattern holds for all rows:

• Row 1: \(63 = 9 \times 7\), \(49 = 7 \times 7\), \(77 = 11 \times 7\) (Factor k=7)
• Row 2: \(99 = 9 \times 11\), \(77 = 7 \times 11\), \(121 = 11 \times 11\) (Factor k=11)
• Row 3: \(135 = 9 \times 15\), \(105 = 7 \times 15\), \(165 = 11 \times 15\) (Factor k=15)

The pattern is consistent across all rows with factors 7, 11, and 15 respectively.

Conclusion

The missing figure in the matrix is 77.

Let's check the given options. The number 77 is one of the options provided.

Revision Table: Matrix Pattern Summary

Additional Information: Matrix Reasoning and Number Patterns

Matrix reasoning questions test your ability to find patterns in a grid of numbers or figures. These patterns can be based on various mathematical operations or logical sequences.

Common types of patterns in number matrices include:

• Arithmetic progression (constant difference between numbers in a row, column, or diagonal).
• Geometric progression (constant ratio between numbers).
• Relationships between numbers in a row (e.g., first + second = third, or operations involving the numbers).
• Relationships between numbers in a column.
• Relationships involving the sum, product, or difference of numbers in a row or column.
• Patterns based on multiplication or division by a common factor or sequence of factors.

In this specific problem, we identified a pattern based on multiplication by a row-specific factor (k), applied to a base set of numbers (9, 7, 11). Another valid pattern found was that each column forms an arithmetic progression.

• Column 1: 63, 99, 135 (difference 36)
• Column 2: 49, 77, 105 (difference 28)
• Column 3: ?, 121, 165 (difference 44). This leads to \(121 - 44 = 77\) for the missing number.

Practicing different types of matrix problems helps in quickly identifying the underlying logic and solving for the missing element.