Questions: Matrices
Matrices26 min readLesson 5 of 6
Q1. What is the order of a matrix with 3 rows and 4 columns, and how many entries does it have?
Answer (English): The order is written rows by columns, so it is 3 by 4. The number of entries is the product of the two numbers, which is 12. An entry is located by its row number and column number, so the entry in row 2, column 3 exists and the entry in row 4, column 1 does not.
Hinglish: Order rows phir columns likha jaata hai, isliye 3 by 4. Entries = 3 x 4 = 12. Entry ka address row aur column se hota hai, isliye row 4 wali entry nahi hogi.
Q2. When are two matrices equal?
Answer (English): Two matrices are equal when they have the same order and every corresponding entry is equal. If the orders differ, the matrices cannot be equal even if the listed numbers look the same. Equal matrices give one equation per entry, which is how missing values are found.
Hinglish: Equality ke liye order same ho aur har entry match kare. Order alag hua to matrices equal nahi ho sakte, numbers same dikhein tab bhi. Equal matrices se har entry ka ek equation milta hai.
Q3. Name the type of matrix that has equal diagonal entries and zeros elsewhere.
Answer (English): That is a scalar matrix. It is a special diagonal matrix in which all diagonal entries are the same number. When that common number is 1, the matrix is the identity matrix. The zero matrix is the case where the common number is 0.
Hinglish: Scalar matrix hota hai jisme diagonal ke saare entries equal hote hain aur baaki 0. Common number 1 ho to identity matrix kehte hain. Common number 0 ho to zero matrix.
Q4. If A is 2 by 3 and B is 3 by 4, what are the orders of AB and BA?
Answer (English): The product AB exists because the inner orders match (3 equals 3), and the order is the outer pair, so 2 by 4. The product BA does not exist, because the columns of B (4) do not match the rows of A (2).
Hinglish: AB ka order inner numbers match hone se outer pair 2 by 4 hota hai. BA possible nahi hai kyunki B ke 4 columns aur A ke 2 rows match nahi karte.
Q5. Find the transpose of A = [[1, 2, 3], [4, 5, 6]].
Answer (English): The matrix A has order 2 by 3. The transpose swaps rows and columns, so it has order 3 by 2 and reads [[1, 4], [2, 5], [3, 6]]. The first row of the transpose is the first column of A, and the pattern continues for the other rows.
Hinglish: A ka order 2 by 3 hai. Transpose me rows aur columns swap hote hain, isliye order 3 by 2 aur matrix [[1, 4], [2, 5], [3, 6]] banti hai.
Q6. Show that A = [[1, 2], [2, 1]] is symmetric.
Answer (English): Transpose A by swapping rows and columns, which gives [[1, 2], [2, 1]]. This is exactly A, so the matrix equals its transpose and is symmetric. Checking one off-diagonal pair confirms the pattern: the entry in row 1, column 2 equals the entry in row 2, column 1.
Hinglish: Transpose lene par wahi matrix milti hai, isliye A' = A aur matrix symmetric hai. Off-diagonal pair bhi equal hai, entry (1, 2) = entry (2, 1).
Q7. For A = [[1, 2], [3, 4]], compute A plus A transpose.
Answer (English): The transpose is [[1, 3], [2, 4]]. Adding entry by entry gives [[2, 5], [5, 8]]. The result is symmetric, which is expected because the sum of any matrix and its transpose always has the property that the entry in row i, column j equals the entry in row j, column i.
Hinglish: Transpose [[1, 3], [2, 4]] banti hai. Add karne par [[2, 5], [5, 8]] milta hai, jo symmetric hai kyunki A + A' hamesha symmetric hota hai.
Q8. If A = [[2, 0], [0, 2]] and B = [[1, 1], [1, 1]], find AB and say whether multiplication is commutative here.
Answer (English): Multiplying, AB = [[2, 2], [2, 2]]. Also BA = [[2, 2], [2, 2]], so for these particular matrices the two products agree. Agreement here does not prove a general rule, because the earlier example of [[1, 2], [3, 4]] with [[5, 6], [7, 8]] gave different products.
Hinglish: AB aur BA dono [[2, 2], [2, 2]] aate hain, isliye is example me product same hai. Par yeh general rule nahi hai, kyunki doosre matrices par products alag aate hain.
Q9. Write the symmetric and skew symmetric parts of A = [[2, 4], [6, 8]].
Answer (English): The transpose is [[2, 6], [4, 8]]. The symmetric part is half of A plus A transpose, which is half of [[4, 10], [10, 16]], that is [[2, 5], [5, 8]]. The skew part is half of A minus A transpose, which is half of [[0, minus 2], [2, 0]], that is [[0, minus 1], [1, 0]]. Adding the two parts returns A.
Hinglish: Symmetric part = (A + A')/2 = [[2, 5], [5, 8]], aur skew part = (A - A')/2 = [[0, -1], [1, 0]]. Dono ko jodne par original matrix wapas aata hai.
Q10. What condition must a square matrix satisfy to be invertible?
Answer (English): A square matrix A is invertible when there exists a square matrix B of the same order such that AB = BA = I, where I is the identity matrix of that order. The matrix B is written A inverse. A matrix without such a partner is called singular, and the next chapter links this to the determinant being zero.
Hinglish: Invertible matrix ke liye same order ka matrix B chahiye jisse AB = BA = I. Tab B ko A inverse likhte hain. Aisa partner na mile to matrix singular hoti hai, aur next chapter me determinant se iska link aata hai.
Marks pattern on this chapter
One mark items ask for order, entry values, symmetry checks or the truth of a matrix identity. Two and three mark items ask for a product, a transpose based decomposition or a short verification. Five mark items combine a product with an equation such as A squared minus kA plus cI equals O. Write the matrix multiplication in one line per row, keep the order written above the answer, and finish with the conclusion sentence. That structure matches the marking scheme step by step.
Quiz below: five multiple choice questions on matrix orders, products, transposes, symmetry and invertibility. Attempt them after revising the formula sheet and the operation rules.