
Without expanding, evaluate the following determinants:
(a)
(b) (c)
Answer
492k+ views
1 likes
Hint:
Here, we need to find the values of the given determinants. We will use the properties of determinants to evaluate the value of the determinants without expanding.
Complete step by step solution:
We will use the properties of determinants to evaluate the given determinants.
(a)
We know that if any two rows or columns of the determinant are added, then the value of the determinant remains the same.
Rewriting column 3 as a sum of column 2 and column 3, we get
Taking common from column 3, we get
If any two rows or columns of a determinant are equal, then its value is zero.
Since column 1 and column 3 are equal, we get
Multiplying the terms, we get
Thus, we get the value of the determinant as 0.
(b)
If any row or column of a determinant is multiplied by a scalar quantity, then the value of the determinant is also multiplied by the same scalar quantity.
Multiplying column 2 by the scalar quantity , we get
Multiplying the terms, we get
We know that if any two columns or rows of a determinant are equal, then its value is zero.
Since column 2 and column 3 are equal, we get
Dividing both sides by , we get
Thus, we get the value of the determinant as 0.
(c)
Rewriting column 3 as a difference of column 3 and 9 times column 2, that is , we get
Multiplying the terms, we get
Subtracting the terms, we get
We know that if any two columns or rows of a determinant are equal, then its value is zero.Since column 1 and column 3 are equal, we get
Thus, we get the value of the determinant as 0.
Note:
We took common from column 3 of to get . This is because of the property of scalar multiplication of a determinant. Multiplying any row or column by a scalar multiplies the value of the determinant by the same scalar quantity.
We need to also remember the properties of determinants:
1) If any two rows or columns of the determinant are added, then the value of the determinant remains the same.
2) If any two columns or rows of a determinant are equal, then its value is zero.
3) If any row or column of a determinant is multiplied by a scalar quantity, then the value of the determinant is also multiplied by the same scalar quantity.
Here, we need to find the values of the given determinants. We will use the properties of determinants to evaluate the value of the determinants without expanding.
Complete step by step solution:
We will use the properties of determinants to evaluate the given determinants.
(a)
We know that if any two rows or columns of the determinant are added, then the value of the determinant remains the same.
Rewriting column 3 as a sum of column 2 and column 3, we get
Taking
If any two rows or columns of a determinant are equal, then its value is zero.
Since column 1 and column 3 are equal, we get
Multiplying the terms, we get
Thus, we get the value of the determinant
(b)
If any row or column of a determinant is multiplied by a scalar quantity, then the value of the determinant is also multiplied by the same scalar quantity.
Multiplying column 2 by the scalar quantity
Multiplying the terms, we get
We know that if any two columns or rows of a determinant are equal, then its value is zero.
Since column 2 and column 3 are equal, we get
Dividing both sides by
Thus, we get the value of the determinant
(c)
Rewriting column 3 as a difference of column 3 and 9 times column 2, that is
Multiplying the terms, we get
Subtracting the terms, we get
We know that if any two columns or rows of a determinant are equal, then its value is zero.Since column 1 and column 3 are equal, we get
Thus, we get the value of the determinant
Note:
We took
We need to also remember the properties of determinants:
1) If any two rows or columns of the determinant are added, then the value of the determinant remains the same.
2) If any two columns or rows of a determinant are equal, then its value is zero.
3) If any row or column of a determinant is multiplied by a scalar quantity, then the value of the determinant is also multiplied by the same scalar quantity.
Latest Vedantu courses for you
Grade 11 Science PCM | CBSE | SCHOOL | English
CBSE (2025-26)
School Full course for CBSE students
₹41,848 per year
Recently Updated Pages
Master Class 11 Economics: Engaging Questions & Answers for Success

Master Class 11 Accountancy: Engaging Questions & Answers for Success

Master Class 11 English: Engaging Questions & Answers for Success

Master Class 11 Social Science: Engaging Questions & Answers for Success

Master Class 11 Physics: Engaging Questions & Answers for Success

Master Class 11 Biology: Engaging Questions & Answers for Success

Trending doubts
How many moles and how many grams of NaCl are present class 11 chemistry CBSE

How do I get the molar mass of urea class 11 chemistry CBSE

Plants which grow in shade are called A Sciophytes class 11 biology CBSE

A renewable exhaustible natural resource is A Petroleum class 11 biology CBSE

In which of the following gametophytes is not independent class 11 biology CBSE

Find the molecular mass of Sulphuric Acid class 11 chemistry CBSE
