If $A = \begin {bmatrix}
\alpha & 0 \\
1 & 1 \end {bmatrix}$ and $B= \begin {bmatrix}
1 & 0 \\
5 & 1 \end {bmatrix}$ then value of $\alpha$ for which $A^2=B$ is
- 1
- 2
- 4
- no real values
The Correct Option is D
Approach Solution - 1
\alpha & 0 \\
1 & 1 \end {bmatrix}$ and B= $\begin {bmatrix}
1 & 0 \\
5 & 1 \end {bmatrix}$
$\Rightarrow \ \ A^2\begin {bmatrix}
\alpha & 0 \\
1 & 1 \end {bmatrix} \begin {bmatrix}
\alpha & 0 \\
1 & 1 \end {bmatrix} = \begin {bmatrix}
\alpha^2 & 0 \\
\alpha+1 & 1 \end {bmatrix}$
Also, given, A$^2$=B
$\Rightarrow \ \ \begin {bmatrix}
\alpha^2 & 0 \\
\alpha+1 & 1 \end {bmatrix} = \begin {bmatrix}
1 & 0 \\
5 & 1 \end {bmatrix}$
$\Rightarrow \ \ \alpha^2=1 \ and \ \alpha+1=5$
Which is not possible at the same time.
$\therefore $ No real values of a exists.
Approach Solution -2
Ans. Which is not possible at the same time. ∴ No real values of α exist.
A rectangular table or array of letters, numbers, or phrases that are organized in columns and rows is called a matrix. A matrix has a size of 2x3 or less since there are two rows and three columns.
The reason Matrix is regarded as one of the most potent tools in mathematics is that it breaks down and simplifies difficult tasks using simple approaches. Matrix has developed as a result of ongoing efforts to identify efficient and compact strategies for solving linear equation systems. The operations of the matrix's notations are used in areas of business including cost estimating, budgeting, and sales prediction. Additionally, the Matrix transcends our screen and is employed for real-world tasks like rotation, reflection, and magnification.
Square Matrix A matrix with the number of rows and columns is equal. A m x n matrix will be square if m = n, and it will be known as the square matrix of order ‘n’. For example, The square matrix of 3 is:
matrix of 3 is:
| 3 | -1 | 0 |
| 3/2 | √3/2 | 1 |
| 4 | 3 | -1 |
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Concepts Used:
Matrices
Matrix:
A matrix is a rectangular array of numbers, variables, symbols, or expressions that are defined for the operations like subtraction, addition, and multiplications. The size of a matrix is determined by the number of rows and columns in the matrix.
The basic operations that can be performed on matrices are:
- Addition of Matrices - The addition of matrices addition can only be possible if the number of rows and columns of both the matrices are the same.
- Subtraction of Matrices - Matrices subtraction is also possible only if the number of rows and columns of both the matrices are the same.
- Scalar Multiplication - The product of a matrix A with any number 'c' is obtained by multiplying every entry of the matrix A by c, is called scalar multiplication.
- Multiplication of Matrices - Matrices multiplication is defined only if the number of columns in the first matrix and rows in the second matrix are equal.
- Transpose of Matrices - Interchanging of rows and columns is known as the transpose of matrices.