Question Video: Evaluating the Determinant of a 3 × 3 Matrix by Using the Properties of the Determinant | Nagwa Question Video: Evaluating the Determinant of a 3 × 3 Matrix by Using the Properties of the Determinant | Nagwa

# Question Video: Evaluating the Determinant of a 3 × 3 Matrix by Using the Properties of the Determinant Mathematics • First Year of Secondary School

## Join Nagwa Classes

If 𝐴 is a matrix of order 3 × 3 such that det (𝐴) = 2, find det (2𝐴).

01:37

### Video Transcript

If 𝐴 is a matrix of order three by three such that the determinant of 𝐴 is two, find the determinant of two 𝐴.

We recall the property of determinants which says that for any square matrix 𝐴 of order 𝑛 by 𝑛 and scalar 𝑘, the determinant of 𝑘𝐴 equals 𝑘 to the power of 𝑛 times the determinant of 𝐴. In this question, we are told matrix 𝐴 has order three by three, which means it has three rows and three columns. We are also told that the determinant of matrix 𝐴 is two.

To answer this question, we need to find the determinant of two 𝐴. Since 𝐴 is a three-by-three matrix, 𝑛 equals three. We will be using a scalar 𝑘 of two. Therefore, according to the property of determinants, by using 𝑛 equals three and 𝑘 equals two, we find that the determinant of two 𝐴 is two to the power of three times the determinant of 𝐴. That is eight times the determinant of 𝐴. And the determinant of 𝐴 is two. Therefore, the determinant of two 𝐴 equals the product of eight and two, which is 16.

In conclusion, given a three-by-three matrix 𝐴 with a determinant of two, the determinant of two 𝐴 is 16.

## Join Nagwa Classes

Attend live sessions on Nagwa Classes to boost your learning with guidance and advice from an expert teacher!

• Interactive Sessions
• Chat & Messaging
• Realistic Exam Questions