Evaluate the determinants in Exercises 1 and 2.
\(\begin{vmatrix}2&4\\-5&-1\end{vmatrix}\)
Evaluate the determinants in Exercises 1 and 2.
\(\begin{vmatrix}2&4\\-5&-1\end{vmatrix}\)
Solution and Explanation
\(\begin{vmatrix}2&4\\-5&-1\end{vmatrix}=\) = 2(−1) − 4(−5) = − 2 + 20 = 18
Top Questions on Determinants
- If $ A $ is a $ 2 \times 2 $ matrix and $ |A| = 4 $, then $ |A^{-1}| $ is:
- CUET (UG) - 2025
- Mathematics
- Determinants
- If \[ A = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 5 & 0 \\ 0 & 0 & -2 \end{bmatrix}, \] then \( |A| \) is:
- CBSE CLASS XII - 2025
- Mathematics
- Determinants
- If \( \left| \begin{matrix} 2x & 5 \\ 12 & x \end{matrix} \right| = \left| \begin{matrix} 6 & -5 \\ 4 & 3 \end{matrix} \right| \), then the value of x is:
- CBSE CLASS XII - 2025
- Mathematics
- Determinants
- If \[ A = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 5 & 0 \\ 0 & 0 & -2 \end{bmatrix}, \] then \( |A| \) is:
- CBSE CLASS XII - 2025
- Mathematics
- Determinants
- If $\left| \begin{array}{ccc} -1 & 2 & 4 \\ 1 & x & 1 \\ 0 & 3 & 3x \end{array} \right| = -57$, the product of the possible values of $x$ is:
- CBSE CLASS XII - 2025
- Mathematics
- Determinants
Questions Asked in CBSE CLASS XII exam
- Determine those values of $x$ for which $f(x) = \frac{2}{x} - 5$, $x \ne 0$ is increasing or decreasing.
- CBSE CLASS XII - 2025
- Functions
- Agro-startups and rural enterprises are becoming increasingly vital for rural development in India. They address the critical challenges faced by rural sector in India. These ventures bring in technological innovations such as precision farming, IT-based monitoring system and data-driven decision making tools that enhance productivity and resource efficiency. By streamlining supply chains and creating direct market linkages, they ensure better pricing for farmers and reduce dependency on traditional intermediaries. Such enterprises contribute actively to employment generation in the rural areas. The government has demonstrated a continuous policy and financial support for agri-startups and entrepreneurs through various programs like RKVY-RAFTAAR, the Agri- infrastructure fund and Agri-SURE.
On the basis of given text and common understanding, answer the following questions:- CBSE CLASS XII - 2025
- Rural Development
Read the following text carefully:
Union Food and Consumer Affairs Minister said that the Central Government has taken many proactive steps in the past few years to control retail prices of food items. He said that the government aims to keep inflation under control without compromising the country’s economic growth. Retail inflation inched up to a three-month high of 5.55% in November 2023 driven by higher food prices. Inflation has been declining since August 2023, when it touched 6.83%. 140 new price monitoring centres had been set up by the Central Government to keep a close watch on wholesale and retail prices of essential commodities. The Government has banned the export of many food items like wheat, broken rice, non-basmati white rice, onions etc. It has also reduced import duties on edible oils and pulses to boost domestic supply and control price rise. On the basis of the given text and common understanding,
answer the following questions:- CBSE CLASS XII - 2025
- Government Budget and the Economy
- Distinguish between Physical Capital and Human Capital.
- CBSE CLASS XII - 2025
- Human Capital Formation
- “In the recent Budget the Government allocated funds for construction of new metro lines in smart cities to improve public transportation and reduce pollution.”
Identify the nature of government expenditure \& its objective.- CBSE CLASS XII - 2025
- Government Budget and the Economy
Concepts Used:
Determinants
Definition of Determinant
A determinant can be defined in many ways for a square matrix.
The first and most simple way is to formulate the determinant by taking into account the top-row elements and the corresponding minors. Take the first element of the top row and multiply it by its minor, then subtract the product of the second element and its minor. Continue to alternately add and subtract the product of each element of the top row with its respective min or until all the elements of the top row have been considered.
For example let us consider a 1×1 matrix A.
A=[a1…….an]
Read More: Properties of Determinants
Second Method to find the determinant:
The second way to define a determinant is to express in terms of the columns of the matrix by expressing an n x n matrix in terms of the column vectors.
Consider the column vectors of matrix A as A = [ a1, a2, a3, …an] where any element aj is a vector of size x.
Then the determinant of matrix A is defined such that
Det [ a1 + a2 …. baj+cv … ax ] = b det (A) + c det [ a1+ a2 + … v … ax ]
Det [ a1 + a2 …. aj aj+1… ax ] = – det [ a1+ a2 + … aj+1 aj … ax ]
Det (I) = 1
Where the scalars are denoted by b and c, a vector of size x is denoted by v, and the identity matrix of size x is denoted by I.
Read More: Minors and Cofactors
We can infer from these equations that the determinant is a linear function of the columns. Further, we observe that the sign of the determinant can be interchanged by interchanging the position of adjacent columns. The identity matrix of the respective unit scalar is mapped by the alternating multi-linear function of the columns. This function is the determinant of the matrix.