Question

Write any 3 properties of Determinants.

Answer 1

Important Properties of Determinants

1. Reflection Property:

The determinant remains unaltered if its rows are changed into columns and the columns into rows. This is known as the property of reflection.

2. All-zero Property:

If all the elements of a row (or column) are zero, then the determinant is zero.

3. Proportionality (Repetition) Property:

If the all elements of a row (or column) are proportional (identical) to the elements of some other row (or column), then the determinant is zero.

4. Switching Property:

The interchange of any two rows (or columns) of the determinant changes its sign.

5. Scalar Multiple Property:

If all the elements of a row (or column) of a determinant are multiplied by a non-zero constant, then the determinant gets multiplied by the same constant.

Answer 2

Important Properties of Determinants

1. ʀᴇғʟᴇᴄᴛɪᴏɴ ᴘʀᴏᴘᴇʀᴛʏ:

ᴛʜᴇ ᴅᴇᴛᴇʀᴍɪɴᴀɴᴛ ʀᴇᴍᴀɪɴs ᴜɴᴀʟᴛᴇʀᴇᴅ ɪғ ɪᴛs ʀᴏᴡs ᴀʀᴇ ᴄʜᴀɴɢᴇᴅ ɪɴᴛᴏ ᴄᴏʟᴜᴍɴs ᴀɴᴅ ᴛʜᴇ ᴄᴏʟᴜᴍɴs ɪɴᴛᴏ ʀᴏᴡs. ᴛʜɪs ɪs ᴋɴᴏᴡɴ ᴀs ᴛʜᴇ ᴘʀᴏᴘᴇʀᴛʏ ᴏғ ʀᴇғʟᴇᴄᴛɪᴏɴ.

2. ᴀʟʟ-ᴢᴇʀᴏ ᴘʀᴏᴘᴇʀᴛʏ:

ɪғ ᴀʟʟ ᴛʜᴇ ᴇʟᴇᴍᴇɴᴛs ᴏғ ᴀ ʀᴏᴡ (ᴏʀ ᴄᴏʟᴜᴍɴ) ᴀʀᴇ ᴢᴇʀᴏ, ᴛʜᴇɴ ᴛʜᴇ ᴅᴇᴛᴇʀᴍɪɴᴀɴᴛ ɪs ᴢᴇʀᴏ.

3. ᴘʀᴏᴘᴏʀᴛɪᴏɴᴀʟɪᴛʏ (ʀᴇᴘᴇᴛɪᴛɪᴏɴ) ᴘʀᴏᴘᴇʀᴛʏ:

ɪғ ᴛʜᴇ ᴀʟʟ ᴇʟᴇᴍᴇɴᴛs ᴏғ ᴀ ʀᴏᴡ (ᴏʀ ᴄᴏʟᴜᴍɴ) ᴀʀᴇ ᴘʀᴏᴘᴏʀᴛɪᴏɴᴀʟ (ɪᴅᴇɴᴛɪᴄᴀʟ) ᴛᴏ ᴛʜᴇ ᴇʟᴇᴍᴇɴᴛs ᴏғ sᴏᴍᴇ ᴏᴛʜᴇʀ ʀᴏᴡ (ᴏʀ ᴄᴏʟᴜᴍɴ), ᴛʜᴇɴ ᴛʜᴇ ᴅᴇᴛᴇʀᴍɪɴᴀɴᴛ ɪs ᴢᴇʀᴏ.

4. sᴡɪᴛᴄʜɪɴɢ ᴘʀᴏᴘᴇʀᴛʏ:

ᴛʜᴇ ɪɴᴛᴇʀᴄʜᴀɴɢᴇ ᴏғ ᴀɴʏ ᴛᴡᴏ ʀᴏᴡs (ᴏʀ ᴄᴏʟᴜᴍɴs) ᴏғ ᴛʜᴇ ᴅᴇᴛᴇʀᴍɪɴᴀɴᴛ ᴄʜᴀɴɢᴇs ɪᴛs sɪɢɴ.

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